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The discriminant of a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d0332001.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d0332002.png" />, whose roots are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d0332003.png" /> is the product
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{{MSC|08|11,12,13,16}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d0332004.png" /></td> </tr></table>
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The notion of ''discriminant'' can have different meanings, depending on the context.
  
The discriminant vanishes if and only if the polynomial has multiple roots. The discriminant is symmetric with respect to the roots of the polynomial and may therefore be expressed in terms of the coefficients of this polynomial.
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====Discriminant of a [[polynomial]]====
  
The discriminant of a quadratic polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d0332005.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d0332006.png" />; the discriminant of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d0332007.png" /> (the roots of which can be computed by the [[Cardano formula|Cardano formula]]) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d0332008.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d0332009.png" /> is a polynomial over a field of characteristic zero, then
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Cf. {{Cite|Ku}}, {{Cite|La}}, {{Cite|Wa}}.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320010.png" /></td> </tr></table>
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The discriminant of a polynomial $f(x)=a_0 x^n+a_1 x^{n-1}+\cdots + a_n$, $a_0 \ne 0$, whose roots are $\def\a{ {\alpha}}\a_1,\dots,\a_n$ is the product
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320011.png" /> is the [[Resultant|resultant]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320012.png" /> and its derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320013.png" />. The derivative of a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320014.png" /> with coefficients from an arbitrary field is the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320015.png" />.
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$$D(f) = a_0^{2n-2} \prod_{n\ge i > j \ge 1}(\a_i - \a_j)^2.$$
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The discriminant vanishes if and only if the polynomial has multiple roots. The discriminant is symmetric with respect to the roots of the polynomial and may therefore be expressed in terms of the coefficients of this polynomial.
  
====References====
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The discriminant of a quadratic polynomial $ax^2+bx+c$ is $b^2-4ac$; the discriminant of the polynomial $x^3+px+q$ (the roots of which can be computed by the
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "Higher algebra" , MIR  (1972) (Translated from Russian)</TD></TR></table>
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[[Cardano formula|Cardano formula]]) is $-27 q^2-4p^3$. If $f(x)$ is a polynomial over a field of characteristic zero, then
  
 +
$$D(f) = (-1)^{n(n-1)/2}a_0^{-1} R(f,f')$$
 +
where $R(f,f')$ is the
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[[Resultant|resultant]] of $f(x)$ and its derivative $f'(x)$. The derivative of a polynomial $f(x) = a_0 x^n + a_1 x^{n-1}+ \cdots + a_n$ with coefficients from an arbitrary field is the polynomial $n a_0 x^{n-1} + (n-1)a_1 x^{n-2}+ \cdots + a_{n-1}$.
  
  
====Comments====
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====Discriminant of a [[sesquilinear form]]====
  
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Cf. {{Cite|Bo}}, {{Cite|Di}}
  
====References====
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The discriminant of a form $f$ sesquilinear with respect to an automorphism $\def\s{ {\sigma}}$ in a basis $(v) = \{v_1,\dots,v_n\}$ is the element of the ring $A$ equal to
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.L. van der Waerden,   "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR></table>
 
  
The discriminant of a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320016.png" /> sesquilinear with respect to an automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320017.png" /> in a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320018.png" /> is the element of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320019.png" /> equal to
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$$D_f(v_1,\dots,v_n) = {\rm det}\begin{pmatrix}
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f(v_1,v_1) & \cdots &f(v_1,v_n)\\
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\vdots &\ddots &\vdots \\
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f(v_n,v_1)&\cdots&f(v_n,v_n)\end{pmatrix},\label{*}$$
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where $(v)$ is a fixed basis of a free $A$-module $E$ of finite rank over the commutative ring $A$ (with a unit element). If $(w) = \{w_1,\dots,w_n\}$ is another basis in $E$ and if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$C=\begin{pmatrix}
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c_{1,1} & \cdots &c_{1,n}\\
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\vdots &\ddots &\vdots \\
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c_{n,1}&\cdots&c_{n,n}\end{pmatrix}$$
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is the transition matrix from $(v)$ to $(w)$, then
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320021.png" /> is a fixed basis of a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320022.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320023.png" /> of finite rank over the commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320024.png" /> (with a unit element). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320025.png" /> is another basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320026.png" /> and if
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$$D_f(w_1,\dots,w_n) = (\det C)(\det C)^\s D_f(v_1,\dots,v_n).$$
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If $A$ has no zero divisors, then for $f$ not to be degenerate it is necessary and sufficient that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320027.png" /></td> </tr></table>
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$$D_f(v_1,\dots,v_n) \ne 0.$$
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If $v_1,\dots,v_n$ are $n$ elements arbitrarily chosen from $E$, then the element $D_f(v_1,\dots,v_n)$ of $A$ defined by (1) is called the discriminant of $f$ with respect to the system $v_1,\dots,v_n$. Let $A$ have no zero divisors and let $f$ be a non-degenerate sesquilinear form. Then, for a system of elements $v_1,\dots,v_n$ from $E$ to be free it is necessary and sufficient that $D_f(v_1,\dots,v_n) \ne 0$. Here, $v_1,\dots,v_n$ form a basis in $E$ if and only if $D_f(v_1,\dots,v_n) $ and $D_f(u_1,\dots,u_n) $ are associated in $A$ for some basis $u_1,\dots,u_n $ in $E$.
  
is the transition matrix from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320028.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320029.png" />, then
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320030.png" /></td> </tr></table>
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''V.L. Popov''
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320031.png" /> has no zero divisors, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320032.png" /> not to be degenerate it is necessary and sufficient that
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====Discriminant of an [[algebra (ring theory) | algebra]]====
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320033.png" /></td> </tr></table>
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Cf. {{Cite|Ja}}.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320034.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320035.png" /> elements arbitrarily chosen from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320036.png" />, then the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320037.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320038.png" /> defined by (*) is called the discriminant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320039.png" /> with respect to the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320040.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320041.png" /> have no zero divisors and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320042.png" /> be a non-degenerate sesquilinear form. Then, for a system of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320043.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320044.png" /> to be free it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320045.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320046.png" /> form a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320047.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320049.png" /> are associated in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320050.png" /> for some basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320051.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320052.png" />.
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The discriminant of an algebra $A$ is the discriminant of the symmetric bilinear form $\def\Tr{ {\rm Tr}}(x,y)=\Tr(xy)$, where $x,y$ are elements of the finite-dimensional associative algebra $A$ over a field $F$, while $\Tr(a)$ is the principal trace of the element $a\in A$, which is defined as follows: Let $e_1,\dots,e_n$ be some basis of the algebra $A$, let $\Phi=F(\xi_1,\dots,\xi_n)$ be a purely
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[[Transcendental extension|transcendental extension]] of the field $F$ formed with algebraically independent elements $\xi_1,\dots,\xi_n$, and let $\def\Ph{ {\Phi}}A_\Ph = A_F\otimes \Ph$ be the corresponding scalar extension of the algebra $A$. An element $x=\xi_1 e_1+\cdots+\xi_n e_n\in A_\Ph$ is then said to be a generic element of the algebra $A$, while the minimal polynomial (over $\Ph$) of the element $x$ is known as the minimal polynomial of the algebra $A$. Let
  
====References====
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$$g(t,\xi) = t^r -m_1(\xi) t^{r-1} +\cdots + (-1)^r m_r(\xi)$$
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Algebra: Modules. Rings. Forms" , '''2''' , Addison-Wesley  (1975) pp. Chapt.4;5;6  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.A. Dieudonné,  "La géométrie des groups classiques" , Springer  (1955)</TD></TR></table>
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be the minimal polynomial of the algebra $A$; the coefficients $m_i(\xi)$ are in fact polynomials from $F[\xi_1,\dots,\xi_n]$. If $\a = \a_1 e_1+\cdots+\a_n e_n$ ($\a_i \in F$) is an arbitrary element of $A$, then $m_1(\a_1,\dots,\a_n) = \Tr(\a)$ is said to be the principal trace of the element $\a$, $m_r(\a_1,\dots,\a_n) = {\rm N}(\a)$ is said to be its principal norm, while the polynomial $g(t,\a_1,\dots,\a_n)$ is known as its principal polynomial. For a given element $\a\in A$ the coefficients of the principal polynomial are independent of the basis chosen; for this reason the bilinear form $(x,y)$ on $A$ which was mentioned above is defined invariantly, while its discriminant is defined up to a multiplicative factor which is the square of a non-zero element of $F$. The algebra $A$ is separable (cf.
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[[Separable algebra|Separable algebra]]) if and only if its discriminant is non-zero.
  
''V.L. Popov''
 
  
The discriminant of a system of elements of a field is one of the most important constructions in the theory of field extensions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320053.png" /> be a finite [[Extension of a field|extension of a field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320054.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320055.png" />. The mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320056.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320057.png" />:
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''E.N. Kuz'min''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320058.png" /></td> </tr></table>
 
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320060.png" /> is the trace of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320061.png" />, is a symmetric bilinear form on the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320062.png" />, which is regarded as a linear space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320063.png" />. The discriminant of this bilinear form (cf. Discriminant of a form) with respect to a system of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320064.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320065.png" /> is said to be the discriminant of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320066.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320067.png" />. In particular, if the system is a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320068.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320069.png" />, its discriminant is called the discriminant of the basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320070.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320071.png" />. The discriminants of two bases differ by a factor which is the square of some non-zero element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320072.png" />. The discriminant of any basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320073.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320074.png" /> is non-zero if and only if the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320075.png" /> is separable (cf. [[Separable extension|Separable extension]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320076.png" /> is a polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320077.png" /> which is the minimal polynomial of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320078.png" /> from the separable extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320079.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320080.png" /> coincides with the discriminant of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320081.png" />. The definitions above can also be applied to an arbitrary finite-dimensional associative algebra over a field (see 4) below).
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====Discriminant of a [[field]]====
  
In the case of a separable extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320082.png" /> the discriminant of the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320083.png" /> may be calculated by the formula
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Cf. {{Cite|BoSh}}, {{Cite|La2}}, {{Cite|Ja}}, {{Cite|ZaSe}}.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320084.png" /></td> </tr></table>
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The discriminant of a system of elements of a field is one of the most important constructions in the theory of field extensions. Let $K$ be a finite
 +
[[Extension of a field|extension of a field]] $k$ of degree $n$. The mapping from $K\times K $ into $k$:
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320085.png" /> are all different imbeddings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320086.png" /> in a given algebraic closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320087.png" /> which leave <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320088.png" /> fixed.
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$$(x,y)\mapsto \def\Tr{ {\rm Tr}} \Tr(xy),$$
 +
where $x,y\in K$ and $\Tr \a$ is the trace of an element $\a\in K$, is a symmetric bilinear form on the field $K$, which is regarded as a linear space over $k$. The discriminant of this bilinear form (cf. Discriminant of a form) with respect to a system of elements $w_1,\dots,w_m $ from $K$ is said to be the discriminant of the system $w_1,\dots,w_m $ and is denoted by $D(w_1,\dots,w_m) $. In particular, if the system is a basis of $K$ over $k$, its discriminant is called the discriminant of the basis of $K$ over $k$. The discriminants of two bases differ by a factor which is the square of some non-zero element of $k$. The discriminant of any basis of $K$ over $k$ is non-zero if and only if the extension $K/k$ is separable (cf.
 +
[[Separable extension|Separable extension]]). If $f_x(t)$ is a polynomial of degree $m$ which is the minimal polynomial of the element $x$ from the separable extension $K/k$, then $D(1,x,\dots,x^m)$ coincides with the discriminant of the polynomial $f_x(t)$. The definitions above can also be applied to an arbitrary finite-dimensional associative algebra over a field (see 4) below).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320089.png" /> be the field of rational numbers, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320090.png" /> be an algebraic number field and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320091.png" /> be some lattice of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320092.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320093.png" />. Then, for any two bases of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320094.png" /> the values of discriminant are identical, and this common value is known as the discriminant of the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320095.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320096.png" /> coincides with the ring of integers of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320097.png" />, the discriminant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320098.png" /> is simply called the discriminant of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d03320099.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200100.png" />; this quantity is an important characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200101.png" />. For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200102.png" /> permits <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200103.png" /> real and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200104.png" /> complex imbeddings in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200105.png" /> of complex numbers, then
+
In the case of a separable extension $K/k$ the discriminant of the basis $w_1,\dots,w_n $ may be calculated by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200106.png" /></td> </tr></table>
+
$$D(w_1,\dots,w_n) = (\det (\s_i(w_j)))^2, $$
 +
where $\s_1,\dots,\s_n $ are all different imbeddings of $K$ in a given algebraic closure of $k$ which leave $k$ fixed.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200107.png" /> is Dedekind's [[Zeta-function|zeta-function]]; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200108.png" /> is the number of [[Divisor|divisor]] classes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200109.png" /> is the regulator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200110.png" /> (cf. [[Regulator of an algebraic number field|Regulator of an algebraic number field]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200111.png" /> is the number of roots of unity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200112.png" />. By virtue of the estimate
+
Let $k=\Q$ be the field of rational numbers, let $K$ be an algebraic number field and let $M$ be some lattice of rank $n$ in $K$. Then, for any two bases of $M$ the values of discriminant are identical, and this common value is known as the discriminant of the lattice $M$. If $M$ coincides with the ring of integers of the field $K$, the discriminant of $M$ is simply called the discriminant of the field $K$ and is denoted by $D_K$; this quantity is an important characteristic of $K$. For instance, if $K$ permits $s$ real and $2t$ complex imbeddings in the field $\C$ of complex numbers, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200113.png" /></td> </tr></table>
+
$$\lim_{q\to 1+0} (q-1) \def\z{ {\zeta}}\z_k(q) = \frac{2^{s+t}\pi^t R}{m\sqrt{|D_K|}}h,$$
 +
where $\z_k(q)$ is Dedekind's
 +
[[Zeta-function|zeta-function]]; $h$ is the number of
 +
[[Divisor|divisor]] classes, $R$ is the regulator of $K$ (cf.
 +
[[Regulator of an algebraic number field|Regulator of an algebraic number field]]) and $m$ is the number of roots of unity in $K$. By virtue of the estimate
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200114.png" />. For a quadratic field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200115.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200116.png" /> is a square-free rational integer, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200117.png" />, one has the formulas
+
$$|D_K|>(\frac{\pi}{4})^{2t} \frac{1}{2\pi n}e^{2n-1/6n},$$
 +
$\lim_{n\to\infty} | D_K|=\infty$. For a quadratic field $\Q(\sqrt{d})$, where $d$ is a square-free rational integer, $d\ne 1$, one has the formulas
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200118.png" /></td> </tr></table>
+
$$D_K = d \quad {\rm if} d\equiv 1 \pmod 4,$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200119.png" /></td> </tr></table>
+
$$D_K = 4d \quad {\rm if} d\equiv 2\; {\rm or }\; 3 \pmod 4.$$
 +
For a
 +
[[Cyclotomic field|cyclotomic field]] $K=\Q(\def\e{ {\epsilon}}\e)$, where $\e$ is a primitive $p^r$-th root of unity, one has
  
For a [[Cyclotomic field|cyclotomic field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200120.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200121.png" /> is a primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200122.png" />-th root of unity, one has
+
$$D_K = \pm p^{p^{r-1} (pr-r-1)}$$
 +
the minus sign being taken if $p^r = 4$ or $p\equiv 3$ ($\mod 4$), while the plus sign is taken in the other cases.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200123.png" /></td> </tr></table>
+
This definition of the discriminant of a lattice in an algebraic number field may be generalized to the case when $k$ is the field of fractions of a
 +
[[Dedekind ring|Dedekind ring]] $A$, and $K$ is a finite separable extension of $k$ of degree $n$. Let $B$ be the integral closure of the ring $A$ in $K$ and let $\def\fb{ {\mathfrak b}}\fb $ be an arbitrary
 +
[[Fractional ideal|fractional ideal]] in $B$. Then the $A$-module $D(\fb)$ generated by all discriminants of the form $D(w_1,\dots,w_n)$, where $w_1,\dots,w_n$ run through all possible bases of $K$ over $k$ and lying in $\fb$, is called the discriminant of the ideal $\fb$. $D(\fb)$ will then be a fractional ideal of $A$, and the equality $D(\fb) = N(\fb)^2 D(B)$, where $N(\fb)$ is the norm of the ideal $\fb$, is valid. The discriminant $D(B)$ is identical with the norm of the different of the ring $B$ over $A$.
  
the minus sign being taken if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200124.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200125.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200126.png" />), while the plus sign is taken in the other cases.
 
 
This definition of the discriminant of a lattice in an algebraic number field may be generalized to the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200127.png" /> is the field of fractions of a [[Dedekind ring|Dedekind ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200128.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200129.png" /> is a finite separable extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200130.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200131.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200132.png" /> be the integral closure of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200133.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200134.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200135.png" /> be an arbitrary [[Fractional ideal|fractional ideal]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200136.png" />. Then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200137.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200138.png" /> generated by all discriminants of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200139.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200140.png" /> run through all possible bases of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200141.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200142.png" /> and lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200143.png" />, is called the discriminant of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200144.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200145.png" /> will then be a fractional ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200146.png" />, and the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200147.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200148.png" /> is the norm of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200149.png" />, is valid. The discriminant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200150.png" /> is identical with the norm of the different of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200151.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200152.png" />.
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Z.I. Borevich,  I.R. Shafarevich,  "Number theory" , Acad. Press  (1966)  (Translated from Russian)  (German translation: Birkhäuser, 1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Lang,  "Algebraic numbers" , Addison-Wesley  (1964)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N. Jacobson,  "The theory of rings" , Amer. Math. Soc.  (1943)</TD></TR></table>
 
  
 
''V.L. Popov''
 
''V.L. Popov''
  
The discriminant of an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200153.png" /> is the discriminant of the symmetric bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200154.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200155.png" /> are elements of the finite-dimensional associative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200156.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200157.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200158.png" /> is the principal trace of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200159.png" />, which is defined as follows: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200160.png" /> be some basis of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200161.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200162.png" /> be a purely [[Transcendental extension|transcendental extension]] of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200163.png" /> formed with algebraically independent elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200164.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200165.png" /> be the corresponding scalar extension of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200166.png" />. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200167.png" /> is then said to be a generic element of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200168.png" />, while the minimal polynomial (over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200169.png" />) of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200170.png" /> is known as the minimal polynomial of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200171.png" />. Let
+
====Discriminant and Different of a [[global field]]====
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200172.png" /></td> </tr></table>
 
 
 
be the minimal polynomial of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200173.png" />; the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200174.png" /> are in fact polynomials from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200175.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200176.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200177.png" />) is an arbitrary element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200178.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200179.png" /> is said to be the principal trace of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200180.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200181.png" /> is said to be its principal norm, while the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200182.png" /> is known as its principal polynomial. For a given element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200183.png" /> the coefficients of the principal polynomial are independent of the basis chosen; for this reason the bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200184.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200185.png" /> which was mentioned above is defined invariantly, while its discriminant is defined up to a multiplicative factor which is the square of a non-zero element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200186.png" />. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200187.png" /> is separable (cf. [[Separable algebra|Separable algebra]]) if and only if its discriminant is non-zero.
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,  "The theory of rings" , Amer. Math. Soc.  (1943)</TD></TR></table>
 
 
 
''E.N. Kuz'min''
 
 
 
====Comments====
 
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200188.png" /> be a global field (an algebraic number field or a function field in one variable) or a local field, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200189.png" /> be a finite separable field extension. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200190.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200191.png" /> be the rings of integers (principal orders) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200192.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200193.png" />, respectively. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200194.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200195.png" /> is the trace function.
 
 
 
(Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200196.png" /> be a finite-dimensional commutative algebra over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200197.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200198.png" /> an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200199.png" />. Choose a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200200.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200201.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200202.png" />. Then multiplication with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200204.png" />, is given by a certain matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200205.png" />. One now defines, the trace, norm and characteristic polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200206.png" /> as the trace, determinant and characteristic polynomial of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200207.png" />:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200208.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200209.png" /></td> </tr></table>
 
  
The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200210.png" /> is a fractional ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200211.png" />. Its inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200212.png" /> in the group of fractional ideals of the Dedekind ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200213.png" /> is called the different of the field extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200214.png" />, and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200215.png" />. Sometimes (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200216.png" />) it is called the relative different, and the (absolute) different of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200217.png" /> is then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200218.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200219.png" /> is a tower of field extensions one has the chain theorem for differents, also called multiplicativity of differents in a tower:
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Cf. {{Cite|Ja2}}, {{Cite|La2}}, {{Cite|We}}, {{Cite|We2}}.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200220.png" /></td> </tr></table>
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Let $F$ be a global field (an algebraic number field or a function field in one variable) or a local field, and let $E/F$ be a finite separable field extension. Let $A_F$ and $A_E$ be the rings of integers (principal orders) of $F$ and $E$, respectively. Let $\def\fm{ {\mathfrak m}} \fm = \{x\in E \mid \Tr(xA_E)\subset A_F \}$ where $\Tr : E\to F$ is the trace function.
  
The ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200221.png" /> is an integral ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200222.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200223.png" />) and it is related to the discriminant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200224.png" /> of the field extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200225.png" /> by
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(Let $B$ be a finite-dimensional commutative algebra over a field $k$ and $b\in B$ an element of $B$. Choose a basis $x_1,\dots,x_n$ of $B$ over $k$. Then multiplication with $b$, $b\mapsto ba$, is given by a certain matrix $M_b$. One now defines, the trace, norm and characteristic polynomial of $b$ as the trace, determinant and characteristic polynomial of the matrix $M_b$:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200226.png" /></td> </tr></table>
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$$\Tr_{B/k}(b) = \Tr (M_b),\quad N_{B/k} = \det (M_b),$$
  
For the different <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200227.png" /> to be divisible by a prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200228.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200229.png" /> it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200230.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200231.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200232.png" />. This is Dedekind's discriminant theorem. Hence a prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200233.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200234.png" /> is ramified in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200235.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200236.png" /> divides the discriminant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200237.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200238.png" />.
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$$f_{B/k}(b)(X) = \det (X {\rm I_n} - M_b).)$$
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The set $\fm$ is a fractional ideal of $A_E$. Its inverse $\fm^{-1}$ in the group of fractional ideals of the Dedekind ring $A_E$ is called the different of the field extension $E/F$, and is denoted by $\def\cD{ {\mathcal D}} \cD_{E/F}$. Sometimes (if $F\ne \Q$) it is called the relative different, and the (absolute) different of $E$ is then $\cD_{E/\Q}$. If $D/E/F$ is a tower of field extensions one has the chain theorem for differents, also called multiplicativity of differents in a tower:
  
Given an additive subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200239.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200240.png" />, its complementary set (relative to the trace) is defined by
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$$\cD_{D/F} = \cD_{D/E} \cD_{E/F}. $$
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The ideal $\cD_{E/F}$ is an integral ideal of $A_E$ (i.e. $\cD_{E/F}\subset A_E$) and it is related to the discriminant $\cD(A_E)$ of the field extension $E/F$ by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200241.png" /></td> </tr></table>
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$$\cD(A_E) = N_{E/F}\cD_{E/F}.$$
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For the different $\cD_{E/F}$ to be divisible by a prime ideal $\def\fp{ {\mathfrak p}}\fp$ of $E$ it is necessary and sufficient that $\def\fq{ {\mathfrak q}}\fq A = \fp^e \fp_2^{e_2}\cdots \fp_m^{e_m}$ with $e>1$, where $\fq = \fp \cap A_F$. This is Dedekind's discriminant theorem. Hence a prime ideal $\fq$ of $F$ is ramified in $E/F$ if and only if $\fq$ divides the discriminant $\cD(A_E)$ of $E/F$.
  
It is also an additive subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200242.png" />. Thus, the different of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200243.png" /> is the inverse of the complementary set of the ring of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200244.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200245.png" />.
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Given an additive subgroup $L$ of $E$, its complementary set (relative to the trace) is defined by
  
More generally one defines the different of an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200246.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200247.png" /> as the inverse of its complementary set: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200248.png" />. It is again a (fractional) ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200249.png" />. The different of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200250.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200251.png" /> is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200252.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200253.png" /> is the derivative of the characteristic polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200254.png" /> of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200255.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200256.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200257.png" />, then the different <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200258.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200259.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200260.png" /> is an integral basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200261.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200262.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200263.png" />.
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$$L' = \{ x\in E \mid \Tr_{E/F}(xL) \subset A_F\}.$$
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It is also an additive subgroup of $A_E$. Thus, the different of $E/F$ is the inverse of the complementary set of the ring of integers $A_E$ of $E$.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200264.png" /> now be a finite extension of global fields. For each prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200265.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200266.png" /> let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200267.png" /> be the corresponding local field (the completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200268.png" /> with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200269.png" />-adic topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200270.png" />). As before, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200271.png" /> is a prime ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200272.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200273.png" /> is the prime ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200274.png" /> underneath it: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200275.png" />. Then one has for the local and global differents that
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More generally one defines the different of an ideal $\def\fa{ {\mathfrak a}}\fa $ in $A_E$ as the inverse of its complementary set: $\cD(\fa) = (\fa')^{-1}$. It is again a (fractional) ideal of $A_E$. The different of an element $x$ in $E$ is defined as $f'(X)$ where $f'(X)$ is the derivative of the characteristic polynomial $f(X)$ of the element $x$ in $E$. If $\a\in \def\cO{ {\mathcal O}}\cO_E$, then the different $\cD(\a)$ is in $\cD_{E/F}$ and $\{1,\a,\dots,\a^{n-1}\}$ is an integral basis of $A_E$ over $A_F$ if and only if $\cD_{E/F} = \cD(\a)A_E$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200276.png" /></td> </tr></table>
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Let $E/F$ now be a finite extension of global fields. For each prime ideal $\fp$ of $E$ let $E_\fp$ be the corresponding local field (the completion of $E$ with respect to the $\fp$-adic topology on $E$). As before, if $\fp$ is a prime ideal of $E$, $\fq$ is the prime ideal of $F$ underneath it: $\fq=\fp\cap A_F$. Then one has for the local and global differents that
  
where an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200277.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200278.png" /> is identified with its completion in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200279.png" />, and where the statement is supposed to include that all but finitely many of the factors on the right-hand side are 1, i.e. unit ideals (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200280.png" /> for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200281.png" />).
+
$$\cD_{E/F} = \prod_\fp\cD_{E_\fp/F_\fq},$$
 +
where an ideal $\fp'$ of $E$ is identified with its completion in $E_{\fp'}$, and where the statement is supposed to include that all but finitely many of the factors on the right-hand side are 1, i.e. unit ideals ($\cD_{E_\fp/F_\fq} = A_{E_\fp},$ for almost-all $\fp$).
  
Let now <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200282.png" /> be a Dedekind integral domain with quotient field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200283.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200284.png" /> be a central simple algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200285.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200286.png" /> is a finite-dimensional associative algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200287.png" /> with no ideals except <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200288.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200289.png" /> and the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200290.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200291.png" />). Then there is a separable normal extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200292.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200293.png" /> (as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200294.png" />-algebras), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200295.png" /> is the algebra of (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200296.png" />)-matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200297.png" />. (Such an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200298.png" /> is called a splitting field for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200299.png" />.) For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200300.png" /> consider the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200301.png" />. The trace of this matrix is an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200302.png" /> (not just of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200303.png" />); it is called the reduced trace and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200304.png" />. (Its definition is also independent of the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200306.png" />.) Similarly one defines the reduced norm, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200307.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200308.png" />.
+
Let now $R$ be a Dedekind integral domain with quotient field $F$ and let $\def\S{ {\Sigma}}\S$ be a central simple algebra over $F$ (i.e. $\S$ is a finite-dimensional associative algebra over $F$ with no ideals except $0$ and $\S$ and the centre of $\S$ is $F$). Then there is a separable normal extension $E/F$ such that $h: \S\otimes_F E \simeq M_n(E)$ (as $E$-algebras), where $M_n(E)$ is the algebra of ($n\times n$)-matrices over $E$. (Such an $E$ is called a splitting field for $\S$.) For each $x\in \S$ consider the element $h(x\otimes 1) \in M_n(E)$. The trace of this matrix is an element of $F$ (not just of $E$); it is called the reduced trace and is denoted by $\Tr : \S \to F$. (Its definition is also independent of the choice of $E$ and $h$.) Similarly one defines the reduced norm, $\def\rn{ {\rm red N}}\rn :\S\to F$, as $\rn = \det(h(x\otimes 1))$.
  
An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200310.png" />-lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200311.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200312.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200313.png" />-submodule of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200314.png" /> that is finitely generated over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200315.png" /> and is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200316.png" />. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200317.png" />-lattice that is a subring and contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200318.png" /> is called an order. A maximal order is an order that is not contained in any other. These always exist but may be non-unique. (These three definitions hold for any separable associative algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200319.png" /> not just for central simple ones.)
+
An $R$-lattice $\fa$ in $\S$ is an $R$-submodule of $\S$ that is finitely generated over $R$ and is such that $F_\fa = \S$. An $R$-lattice that is a subring and contains $R$ is called an order. A maximal order is an order that is not contained in any other. These always exist but may be non-unique. (These three definitions hold for any separable associative algebra over $F$ not just for central simple ones.)
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200320.png" /> be a maximal order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200321.png" />. The different of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200322.png" /> in this setting is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200323.png" />. The discriminant of a central simple algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200324.png" /> is the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200325.png" />. It does not depend on the choice of the maximal order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033200/d033200326.png" />.
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Let $A$ be a maximal order in $\S$. The different of $A$ in this setting is defined by $\cD(A)^{-1} = \{x\in E \mid \rn (xA)\subset R\}$. The discriminant of a central simple algebra $\S$ is the ideal $\delta = \rn(\cD(A))$. It does not depend on the choice of the maximal order $A$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Jacobson,  "Lectures in abstract algebra" , '''3. Theory of fields and galois theory''' , v. Nostrand  (1964)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S. Lang,  "Algebraic numbers" , Addison-Wesley  (1964)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Weil,  "Basic number theory" , Springer  (1967)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E. Weiss,  "Algebraic number theory" , McGraw-Hill  (1963)  pp. Sects. 4–9</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Bo}}||valign="top"|  N. Bourbaki,  "Elements of mathematics. Algebra: Modules. Rings. Forms", '''2''', Addison-Wesley  (1975)  pp. Chapt.4;5;6  (Translated from French) {{MR|0643362}} {{ZBL|1139.12001}}
 +
|-
 +
|valign="top"|{{Ref|BoSh}}||valign="top"|  Z.I. Borevich,  I.R. Shafarevich,  "Number theory", Acad. Press  (1966)  (Translated from Russian)  (German translation: Birkhäuser, 1966) {{MR|0195803}} {{ZBL|0145.04902}}
 +
|-
 +
|valign="top"|{{Ref|Di}}||valign="top"|  J.A. Dieudonné,  "La géométrie des groups classiques", Springer  (1955){{MR|}} {{ZBL|0221.20056}}
 +
|-
 +
|valign="top"|{{Ref|Ja}}||valign="top"|  N. Jacobson,  "The theory of rings", Amer. Math. Soc.  (1943) {{MR|0008601}} {{ZBL|0060.07302}}
 +
|-
 +
|valign="top"|{{Ref|Ja3}}||valign="top"| N. Jacobson,  "Lectures in abstract algebra", ''3. Theory of fields and galois theory'', v. Nostrand  (1964) {{MR|0008601}} {{ZBL|0060.07302}}
 +
|-
 +
|valign="top"|{{Ref|Ku}}||valign="top"|  A.G. Kurosh,  "Higher algebra", MIR  (1972)  (Translated from Russian) {{MR|0945393}} {{MR|0926059}} {{MR|0778202}} {{MR|0759341}} {{MR|0628003}} {{MR|0384363}} {{ZBL|0237.13001}}
 +
|-
 +
|valign="top"|{{Ref|La}}||valign="top"|  S. Lang,  "Algebra", Addison-Wesley  (1974) {{MR|0783636}} {{ZBL|0712.00001}}
 +
|-
 +
|valign="top"|{{Ref|La2}}||valign="top"| S. Lang,  "Algebraic numbers", Addison-Wesley  (1964) {{MR|0160763}} {{ZBL|0211.38501}}
 +
|-
 +
|valign="top"|{{Ref|Wa}}||valign="top"|  B.L. van der Waerden,  "Algebra", '''1–2''', Springer  (1967–1971)  (Translated from German) {{MR|0263582}} {{ZBL|1032.00001}} {{ZBL|1032.00002}}
 +
|-
 +
|valign="top"|{{Ref|We}}||valign="top"| A. Weil,  "Basic number theory", Springer  (1967) {{MR|0234930}} {{ZBL|0176.33601}}
 +
|-
 +
|valign="top"|{{Ref|We2}}||valign="top"| E. Weiss,  "Algebraic number theory", McGraw-Hill  (1963)  pp. Sects. 4–9 {{MR|0159805}} {{ZBL|0115.03601}}
 +
|-
 +
|valign="top"|{{Ref|ZaSa}}||valign="top"|  O. Zariski,  P. Samuel,  "Commutative algebra", '''1''', Springer  (1975) {{MR|0389876}} {{MR|0384768}} {{ZBL|0313.13001}}
 +
|-
 +
|}

Latest revision as of 08:53, 9 December 2016

2020 Mathematics Subject Classification: Primary: 08-XX Secondary: 11-XX12-XX13-XX16-XX [MSN][ZBL]

The notion of discriminant can have different meanings, depending on the context.

Discriminant of a polynomial

Cf. [Ku], [La], [Wa].

The discriminant of a polynomial $f(x)=a_0 x^n+a_1 x^{n-1}+\cdots + a_n$, $a_0 \ne 0$, whose roots are $\def\a{ {\alpha}}\a_1,\dots,\a_n$ is the product

$$D(f) = a_0^{2n-2} \prod_{n\ge i > j \ge 1}(\a_i - \a_j)^2.$$ The discriminant vanishes if and only if the polynomial has multiple roots. The discriminant is symmetric with respect to the roots of the polynomial and may therefore be expressed in terms of the coefficients of this polynomial.

The discriminant of a quadratic polynomial $ax^2+bx+c$ is $b^2-4ac$; the discriminant of the polynomial $x^3+px+q$ (the roots of which can be computed by the Cardano formula) is $-27 q^2-4p^3$. If $f(x)$ is a polynomial over a field of characteristic zero, then

$$D(f) = (-1)^{n(n-1)/2}a_0^{-1} R(f,f')$$ where $R(f,f')$ is the resultant of $f(x)$ and its derivative $f'(x)$. The derivative of a polynomial $f(x) = a_0 x^n + a_1 x^{n-1}+ \cdots + a_n$ with coefficients from an arbitrary field is the polynomial $n a_0 x^{n-1} + (n-1)a_1 x^{n-2}+ \cdots + a_{n-1}$.


Discriminant of a sesquilinear form

Cf. [Bo], [Di]

The discriminant of a form $f$ sesquilinear with respect to an automorphism $\def\s{ {\sigma}}$ in a basis $(v) = \{v_1,\dots,v_n\}$ is the element of the ring $A$ equal to

$$D_f(v_1,\dots,v_n) = {\rm det}\begin{pmatrix} f(v_1,v_1) & \cdots &f(v_1,v_n)\\ \vdots &\ddots &\vdots \\ f(v_n,v_1)&\cdots&f(v_n,v_n)\end{pmatrix},\label{*}$$ where $(v)$ is a fixed basis of a free $A$-module $E$ of finite rank over the commutative ring $A$ (with a unit element). If $(w) = \{w_1,\dots,w_n\}$ is another basis in $E$ and if

$$C=\begin{pmatrix} c_{1,1} & \cdots &c_{1,n}\\ \vdots &\ddots &\vdots \\ c_{n,1}&\cdots&c_{n,n}\end{pmatrix}$$ is the transition matrix from $(v)$ to $(w)$, then

$$D_f(w_1,\dots,w_n) = (\det C)(\det C)^\s D_f(v_1,\dots,v_n).$$ If $A$ has no zero divisors, then for $f$ not to be degenerate it is necessary and sufficient that

$$D_f(v_1,\dots,v_n) \ne 0.$$ If $v_1,\dots,v_n$ are $n$ elements arbitrarily chosen from $E$, then the element $D_f(v_1,\dots,v_n)$ of $A$ defined by (1) is called the discriminant of $f$ with respect to the system $v_1,\dots,v_n$. Let $A$ have no zero divisors and let $f$ be a non-degenerate sesquilinear form. Then, for a system of elements $v_1,\dots,v_n$ from $E$ to be free it is necessary and sufficient that $D_f(v_1,\dots,v_n) \ne 0$. Here, $v_1,\dots,v_n$ form a basis in $E$ if and only if $D_f(v_1,\dots,v_n) $ and $D_f(u_1,\dots,u_n) $ are associated in $A$ for some basis $u_1,\dots,u_n $ in $E$.


V.L. Popov

Discriminant of an algebra

Cf. [Ja].

The discriminant of an algebra $A$ is the discriminant of the symmetric bilinear form $\def\Tr{ {\rm Tr}}(x,y)=\Tr(xy)$, where $x,y$ are elements of the finite-dimensional associative algebra $A$ over a field $F$, while $\Tr(a)$ is the principal trace of the element $a\in A$, which is defined as follows: Let $e_1,\dots,e_n$ be some basis of the algebra $A$, let $\Phi=F(\xi_1,\dots,\xi_n)$ be a purely transcendental extension of the field $F$ formed with algebraically independent elements $\xi_1,\dots,\xi_n$, and let $\def\Ph{ {\Phi}}A_\Ph = A_F\otimes \Ph$ be the corresponding scalar extension of the algebra $A$. An element $x=\xi_1 e_1+\cdots+\xi_n e_n\in A_\Ph$ is then said to be a generic element of the algebra $A$, while the minimal polynomial (over $\Ph$) of the element $x$ is known as the minimal polynomial of the algebra $A$. Let

$$g(t,\xi) = t^r -m_1(\xi) t^{r-1} +\cdots + (-1)^r m_r(\xi)$$ be the minimal polynomial of the algebra $A$; the coefficients $m_i(\xi)$ are in fact polynomials from $F[\xi_1,\dots,\xi_n]$. If $\a = \a_1 e_1+\cdots+\a_n e_n$ ($\a_i \in F$) is an arbitrary element of $A$, then $m_1(\a_1,\dots,\a_n) = \Tr(\a)$ is said to be the principal trace of the element $\a$, $m_r(\a_1,\dots,\a_n) = {\rm N}(\a)$ is said to be its principal norm, while the polynomial $g(t,\a_1,\dots,\a_n)$ is known as its principal polynomial. For a given element $\a\in A$ the coefficients of the principal polynomial are independent of the basis chosen; for this reason the bilinear form $(x,y)$ on $A$ which was mentioned above is defined invariantly, while its discriminant is defined up to a multiplicative factor which is the square of a non-zero element of $F$. The algebra $A$ is separable (cf. Separable algebra) if and only if its discriminant is non-zero.


E.N. Kuz'min


Discriminant of a field

Cf. [BoSh], [La2], [Ja], [ZaSe].

The discriminant of a system of elements of a field is one of the most important constructions in the theory of field extensions. Let $K$ be a finite extension of a field $k$ of degree $n$. The mapping from $K\times K $ into $k$:

$$(x,y)\mapsto \def\Tr{ {\rm Tr}} \Tr(xy),$$ where $x,y\in K$ and $\Tr \a$ is the trace of an element $\a\in K$, is a symmetric bilinear form on the field $K$, which is regarded as a linear space over $k$. The discriminant of this bilinear form (cf. Discriminant of a form) with respect to a system of elements $w_1,\dots,w_m $ from $K$ is said to be the discriminant of the system $w_1,\dots,w_m $ and is denoted by $D(w_1,\dots,w_m) $. In particular, if the system is a basis of $K$ over $k$, its discriminant is called the discriminant of the basis of $K$ over $k$. The discriminants of two bases differ by a factor which is the square of some non-zero element of $k$. The discriminant of any basis of $K$ over $k$ is non-zero if and only if the extension $K/k$ is separable (cf. Separable extension). If $f_x(t)$ is a polynomial of degree $m$ which is the minimal polynomial of the element $x$ from the separable extension $K/k$, then $D(1,x,\dots,x^m)$ coincides with the discriminant of the polynomial $f_x(t)$. The definitions above can also be applied to an arbitrary finite-dimensional associative algebra over a field (see 4) below).

In the case of a separable extension $K/k$ the discriminant of the basis $w_1,\dots,w_n $ may be calculated by the formula

$$D(w_1,\dots,w_n) = (\det (\s_i(w_j)))^2, $$ where $\s_1,\dots,\s_n $ are all different imbeddings of $K$ in a given algebraic closure of $k$ which leave $k$ fixed.

Let $k=\Q$ be the field of rational numbers, let $K$ be an algebraic number field and let $M$ be some lattice of rank $n$ in $K$. Then, for any two bases of $M$ the values of discriminant are identical, and this common value is known as the discriminant of the lattice $M$. If $M$ coincides with the ring of integers of the field $K$, the discriminant of $M$ is simply called the discriminant of the field $K$ and is denoted by $D_K$; this quantity is an important characteristic of $K$. For instance, if $K$ permits $s$ real and $2t$ complex imbeddings in the field $\C$ of complex numbers, then

$$\lim_{q\to 1+0} (q-1) \def\z{ {\zeta}}\z_k(q) = \frac{2^{s+t}\pi^t R}{m\sqrt{|D_K|}}h,$$ where $\z_k(q)$ is Dedekind's zeta-function; $h$ is the number of divisor classes, $R$ is the regulator of $K$ (cf. Regulator of an algebraic number field) and $m$ is the number of roots of unity in $K$. By virtue of the estimate

$$|D_K|>(\frac{\pi}{4})^{2t} \frac{1}{2\pi n}e^{2n-1/6n},$$ $\lim_{n\to\infty} | D_K|=\infty$. For a quadratic field $\Q(\sqrt{d})$, where $d$ is a square-free rational integer, $d\ne 1$, one has the formulas

$$D_K = d \quad {\rm if} d\equiv 1 \pmod 4,$$

$$D_K = 4d \quad {\rm if} d\equiv 2\; {\rm or }\; 3 \pmod 4.$$ For a cyclotomic field $K=\Q(\def\e{ {\epsilon}}\e)$, where $\e$ is a primitive $p^r$-th root of unity, one has

$$D_K = \pm p^{p^{r-1} (pr-r-1)}$$ the minus sign being taken if $p^r = 4$ or $p\equiv 3$ ($\mod 4$), while the plus sign is taken in the other cases.

This definition of the discriminant of a lattice in an algebraic number field may be generalized to the case when $k$ is the field of fractions of a Dedekind ring $A$, and $K$ is a finite separable extension of $k$ of degree $n$. Let $B$ be the integral closure of the ring $A$ in $K$ and let $\def\fb{ {\mathfrak b}}\fb $ be an arbitrary fractional ideal in $B$. Then the $A$-module $D(\fb)$ generated by all discriminants of the form $D(w_1,\dots,w_n)$, where $w_1,\dots,w_n$ run through all possible bases of $K$ over $k$ and lying in $\fb$, is called the discriminant of the ideal $\fb$. $D(\fb)$ will then be a fractional ideal of $A$, and the equality $D(\fb) = N(\fb)^2 D(B)$, where $N(\fb)$ is the norm of the ideal $\fb$, is valid. The discriminant $D(B)$ is identical with the norm of the different of the ring $B$ over $A$.


V.L. Popov

Discriminant and Different of a global field

Cf. [Ja2], [La2], [We], [We2].

Let $F$ be a global field (an algebraic number field or a function field in one variable) or a local field, and let $E/F$ be a finite separable field extension. Let $A_F$ and $A_E$ be the rings of integers (principal orders) of $F$ and $E$, respectively. Let $\def\fm{ {\mathfrak m}} \fm = \{x\in E \mid \Tr(xA_E)\subset A_F \}$ where $\Tr : E\to F$ is the trace function.

(Let $B$ be a finite-dimensional commutative algebra over a field $k$ and $b\in B$ an element of $B$. Choose a basis $x_1,\dots,x_n$ of $B$ over $k$. Then multiplication with $b$, $b\mapsto ba$, is given by a certain matrix $M_b$. One now defines, the trace, norm and characteristic polynomial of $b$ as the trace, determinant and characteristic polynomial of the matrix $M_b$:

$$\Tr_{B/k}(b) = \Tr (M_b),\quad N_{B/k} = \det (M_b),$$

$$f_{B/k}(b)(X) = \det (X {\rm I_n} - M_b).)$$ The set $\fm$ is a fractional ideal of $A_E$. Its inverse $\fm^{-1}$ in the group of fractional ideals of the Dedekind ring $A_E$ is called the different of the field extension $E/F$, and is denoted by $\def\cD{ {\mathcal D}} \cD_{E/F}$. Sometimes (if $F\ne \Q$) it is called the relative different, and the (absolute) different of $E$ is then $\cD_{E/\Q}$. If $D/E/F$ is a tower of field extensions one has the chain theorem for differents, also called multiplicativity of differents in a tower:

$$\cD_{D/F} = \cD_{D/E} \cD_{E/F}. $$ The ideal $\cD_{E/F}$ is an integral ideal of $A_E$ (i.e. $\cD_{E/F}\subset A_E$) and it is related to the discriminant $\cD(A_E)$ of the field extension $E/F$ by

$$\cD(A_E) = N_{E/F}\cD_{E/F}.$$ For the different $\cD_{E/F}$ to be divisible by a prime ideal $\def\fp{ {\mathfrak p}}\fp$ of $E$ it is necessary and sufficient that $\def\fq{ {\mathfrak q}}\fq A = \fp^e \fp_2^{e_2}\cdots \fp_m^{e_m}$ with $e>1$, where $\fq = \fp \cap A_F$. This is Dedekind's discriminant theorem. Hence a prime ideal $\fq$ of $F$ is ramified in $E/F$ if and only if $\fq$ divides the discriminant $\cD(A_E)$ of $E/F$.

Given an additive subgroup $L$ of $E$, its complementary set (relative to the trace) is defined by

$$L' = \{ x\in E \mid \Tr_{E/F}(xL) \subset A_F\}.$$ It is also an additive subgroup of $A_E$. Thus, the different of $E/F$ is the inverse of the complementary set of the ring of integers $A_E$ of $E$.

More generally one defines the different of an ideal $\def\fa{ {\mathfrak a}}\fa $ in $A_E$ as the inverse of its complementary set: $\cD(\fa) = (\fa')^{-1}$. It is again a (fractional) ideal of $A_E$. The different of an element $x$ in $E$ is defined as $f'(X)$ where $f'(X)$ is the derivative of the characteristic polynomial $f(X)$ of the element $x$ in $E$. If $\a\in \def\cO{ {\mathcal O}}\cO_E$, then the different $\cD(\a)$ is in $\cD_{E/F}$ and $\{1,\a,\dots,\a^{n-1}\}$ is an integral basis of $A_E$ over $A_F$ if and only if $\cD_{E/F} = \cD(\a)A_E$.

Let $E/F$ now be a finite extension of global fields. For each prime ideal $\fp$ of $E$ let $E_\fp$ be the corresponding local field (the completion of $E$ with respect to the $\fp$-adic topology on $E$). As before, if $\fp$ is a prime ideal of $E$, $\fq$ is the prime ideal of $F$ underneath it: $\fq=\fp\cap A_F$. Then one has for the local and global differents that

$$\cD_{E/F} = \prod_\fp\cD_{E_\fp/F_\fq},$$ where an ideal $\fp'$ of $E$ is identified with its completion in $E_{\fp'}$, and where the statement is supposed to include that all but finitely many of the factors on the right-hand side are 1, i.e. unit ideals ($\cD_{E_\fp/F_\fq} = A_{E_\fp},$ for almost-all $\fp$).

Let now $R$ be a Dedekind integral domain with quotient field $F$ and let $\def\S{ {\Sigma}}\S$ be a central simple algebra over $F$ (i.e. $\S$ is a finite-dimensional associative algebra over $F$ with no ideals except $0$ and $\S$ and the centre of $\S$ is $F$). Then there is a separable normal extension $E/F$ such that $h: \S\otimes_F E \simeq M_n(E)$ (as $E$-algebras), where $M_n(E)$ is the algebra of ($n\times n$)-matrices over $E$. (Such an $E$ is called a splitting field for $\S$.) For each $x\in \S$ consider the element $h(x\otimes 1) \in M_n(E)$. The trace of this matrix is an element of $F$ (not just of $E$); it is called the reduced trace and is denoted by $\Tr : \S \to F$. (Its definition is also independent of the choice of $E$ and $h$.) Similarly one defines the reduced norm, $\def\rn{ {\rm red N}}\rn :\S\to F$, as $\rn = \det(h(x\otimes 1))$.

An $R$-lattice $\fa$ in $\S$ is an $R$-submodule of $\S$ that is finitely generated over $R$ and is such that $F_\fa = \S$. An $R$-lattice that is a subring and contains $R$ is called an order. A maximal order is an order that is not contained in any other. These always exist but may be non-unique. (These three definitions hold for any separable associative algebra over $F$ not just for central simple ones.)

Let $A$ be a maximal order in $\S$. The different of $A$ in this setting is defined by $\cD(A)^{-1} = \{x\in E \mid \rn (xA)\subset R\}$. The discriminant of a central simple algebra $\S$ is the ideal $\delta = \rn(\cD(A))$. It does not depend on the choice of the maximal order $A$.

References

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How to Cite This Entry:
Discriminant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Discriminant&oldid=18875
This article was adapted from an original article by I.V. Proskuryakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article