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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c1300801.png" /> be a normal (finite-degree) extension of algebraic number fields with Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c1300802.png" />. Pick a prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c1300803.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c1300804.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c1300805.png" /> be the prime ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c1300806.png" /> under it, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c1300807.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c1300808.png" /> is the ring of integers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c1300809.png" />. There is a unique element
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<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/c/c130/c130080/c13008010.png" /></td> </tr></table>
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Out of 35 formulas, 35 were replaced by TEX code.-->
  
of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008011.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008012.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008013.png" /> integral. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008014.png" />, the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008015.png" />, is the number of elements of the residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008016.png" />. This is the [[Frobenius automorphism|Frobenius automorphism]] (or Frobenius symbol) associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008017.png" />.
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Let $L / K$ be a normal (finite-degree) extension of algebraic number fields with Galois group $\operatorname {Gal}( L / K )$. Pick a prime ideal $\frak P$ of $L$ and let $\mathfrak{p}$ be the prime ideal of $K$ under it, i.e. $\mathfrak { p } = A _ { K } \cap \mathfrak { P }$, where $A _ { K }$ is the ring of integers of $K$. There is a unique element
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008018.png" /> is unramified in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008019.png" />, define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008020.png" /> as the [[conjugacy class]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008021.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008023.png" /> is any prime ideal above <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008024.png" />. This conjugacy class depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008025.png" />.
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\begin{equation*} \sigma _ { \mathfrak { P } } = \left[ \frac { L / K } { \mathfrak { P } } \right] \end{equation*}
  
The weak form of the Chebotarev density theorem says that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008026.png" /> is an arbitrary conjugacy class in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008027.png" />, then the set
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of $\operatorname {Gal}( L / K )$ such that $\sigma _ { \mathfrak { P } } \equiv x ^ { N ( \mathfrak { p } ) } \operatorname { mod } \mathfrak { P }$ for $x \in L$ integral. Here, $N ( \mathfrak{p} )$, the norm of $\mathfrak{p}$, is the number of elements of the residue field $A _ { K } / \mathfrak{p}$. This is the [[Frobenius automorphism|Frobenius automorphism]] (or Frobenius symbol) associated to $\frak P$.
  
<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/c/c130/c130080/c13008028.png" /></td> </tr></table>
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If $\mathfrak{p}$ is unramified in $L / K$, define $F _ { L / K } ( \mathfrak{p} )$ as the [[conjugacy class]] of $\sigma _ { \mathfrak{P} }$ in $\operatorname {Gal}( L / K )$, where $\frak P$ is any prime ideal above $\mathfrak{p}$. This conjugacy class depends only on $\mathfrak{p}$.
  
is infinite and has [[Dirichlet density|Dirichlet density]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008030.png" />.
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The weak form of the Chebotarev density theorem says that if $A$ is an arbitrary conjugacy class in $\operatorname {Gal}( L / K )$, then the set
  
The stonger form specifies in addition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008031.png" /> is regular (see [[Dirichlet density|Dirichlet density]]) and that
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\begin{equation*} P _ { A } = \{ \mathfrak { p } : F _ { L / K}  ( \mathfrak { p } ) = A \} \end{equation*}
  
<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/c/c130/c130080/c13008032.png" /></td> </tr></table>
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is infinite and has [[Dirichlet density|Dirichlet density]] $\# A / n$, where $n = [ L : K ]$.
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008033.png" /> the number of prime ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008034.png" /> with norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130080/c13008035.png" />.
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The stonger form specifies in addition that $P _ { A }$ is regular (see [[Dirichlet density|Dirichlet density]]) and that
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\begin{equation*} N _ { A } = \left( \# \frac { A } { n } + o ( 1 ) \right) x \operatorname { log } x, \end{equation*}
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with $N _ { A } ( x )$ the number of prime ideals in $P _ { A }$ with norm $\leq x$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Narkiewicz,  "Elementary and analytic theory of algebraic numbers" , PWN/Springer  (1990)  pp. Sect. 7.3  (Edition: Second)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.G. Chebotarev,  "Determination of the density of the set of primes corresponding to a given class of permutations"  ''Izv. Akad. Nauk.'' , '''17'''  (1923)  pp. 205–230; 231–250  (In Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.G. Chebotarev,  "Die Bestimmung der Dichtigkeit einer Menge von Primzahlen welche zu einer gegebenen Substitutionsklasse gehören"  ''Math. Ann.'' , '''95'''  (1926)  pp. 191–228</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  W. Narkiewicz,  "Elementary and analytic theory of algebraic numbers" , PWN/Springer  (1990)  pp. Sect. 7.3  (Edition: Second)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  N.G. Chebotarev,  "Determination of the density of the set of primes corresponding to a given class of permutations"  ''Izv. Akad. Nauk.'' , '''17'''  (1923)  pp. 205–230; 231–250  (In Russian)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  N.G. Chebotarev,  "Die Bestimmung der Dichtigkeit einer Menge von Primzahlen welche zu einer gegebenen Substitutionsklasse gehören"  ''Math. Ann.'' , '''95'''  (1926)  pp. 191–228</td></tr></table>

Latest revision as of 17:00, 1 July 2020

Let $L / K$ be a normal (finite-degree) extension of algebraic number fields with Galois group $\operatorname {Gal}( L / K )$. Pick a prime ideal $\frak P$ of $L$ and let $\mathfrak{p}$ be the prime ideal of $K$ under it, i.e. $\mathfrak { p } = A _ { K } \cap \mathfrak { P }$, where $A _ { K }$ is the ring of integers of $K$. There is a unique element

\begin{equation*} \sigma _ { \mathfrak { P } } = \left[ \frac { L / K } { \mathfrak { P } } \right] \end{equation*}

of $\operatorname {Gal}( L / K )$ such that $\sigma _ { \mathfrak { P } } \equiv x ^ { N ( \mathfrak { p } ) } \operatorname { mod } \mathfrak { P }$ for $x \in L$ integral. Here, $N ( \mathfrak{p} )$, the norm of $\mathfrak{p}$, is the number of elements of the residue field $A _ { K } / \mathfrak{p}$. This is the Frobenius automorphism (or Frobenius symbol) associated to $\frak P$.

If $\mathfrak{p}$ is unramified in $L / K$, define $F _ { L / K } ( \mathfrak{p} )$ as the conjugacy class of $\sigma _ { \mathfrak{P} }$ in $\operatorname {Gal}( L / K )$, where $\frak P$ is any prime ideal above $\mathfrak{p}$. This conjugacy class depends only on $\mathfrak{p}$.

The weak form of the Chebotarev density theorem says that if $A$ is an arbitrary conjugacy class in $\operatorname {Gal}( L / K )$, then the set

\begin{equation*} P _ { A } = \{ \mathfrak { p } : F _ { L / K} ( \mathfrak { p } ) = A \} \end{equation*}

is infinite and has Dirichlet density $\# A / n$, where $n = [ L : K ]$.

The stonger form specifies in addition that $P _ { A }$ is regular (see Dirichlet density) and that

\begin{equation*} N _ { A } = \left( \# \frac { A } { n } + o ( 1 ) \right) x \operatorname { log } x, \end{equation*}

with $N _ { A } ( x )$ the number of prime ideals in $P _ { A }$ with norm $\leq x$.

References

[a1] W. Narkiewicz, "Elementary and analytic theory of algebraic numbers" , PWN/Springer (1990) pp. Sect. 7.3 (Edition: Second)
[a2] N.G. Chebotarev, "Determination of the density of the set of primes corresponding to a given class of permutations" Izv. Akad. Nauk. , 17 (1923) pp. 205–230; 231–250 (In Russian)
[a3] N.G. Chebotarev, "Die Bestimmung der Dichtigkeit einer Menge von Primzahlen welche zu einer gegebenen Substitutionsklasse gehören" Math. Ann. , 95 (1926) pp. 191–228
How to Cite This Entry:
Chebotarev density theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chebotarev_density_theorem&oldid=35122
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article