...egree 1. In the case of several variables there are absolutely irreducible polynomials of arbitrarily high degree, for example, any polynomial of the form $f(x_1,
...lynomials of arbitrarily high degree; for example, $x^n+px+p$, where $n>1$ and $p$ is a prime number, is irreducible in $\mathbf Q[x]$ by Eisenstein's cri
3 KB (478 words) - 15:26, 30 December 2018
...]$, then $c(g_1g_2) = c(g_1)c(g_2)$. In particular, a product of primitive polynomials is a primitive polynomial.
...p $E^*$ where $E$ is the extension of $F$ by the roots of $f$, cf [[Galois field structure]].
2 KB (271 words) - 19:29, 2 November 2014
...many elements. The field of rational numbers is contained in every number field.
...re $\alpha$ is a fixed complex number and $H(x)$ and $F(x)$ range over the polynomials with rational coefficients.
2 KB (261 words) - 20:42, 23 November 2023
...,|A|\rangle$, with $\Omega$ the signature of $L$ (cf. [[Model theory|Model theory]]).
...n every $\alpha$-saturated extension of $A$ (cf. also [[Model theory|Model theory]]).
3 KB (454 words) - 20:57, 22 December 2018
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A relation between integral polynomials $ a( x) $
4 KB (700 words) - 18:53, 18 January 2024
in which the coefficients located at equal distances from the beginning and from the end are equal: $a_i=a_{n-i}$. A reciprocal equation of degree $2n$
[[Category:Field theory and polynomials]]
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''finite field''
A field with a finite number of elements. First considered by E. Galois [[#Referenc
4 KB (749 words) - 18:32, 2 November 2014
''of a field $K$''
An algebraic field extension (cf. [[Extension of a field]]) $L$ of $K$ satisfying one of the following equivalent conditions:
2 KB (288 words) - 18:30, 11 April 2016
$#C+1 = 56 : ~/encyclopedia/old_files/data/C027/C.0207580 Cyclotomic polynomials,
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4 KB (652 words) - 05:18, 7 March 2022
A polynomial $f$ with coefficients in a field or a commutative associative ring $K$ with a unit, which is a [[Symmetric f
The symmetric polynomials form the algebra $S(x_1,\ldots,x_n)$ over $K$.
5 KB (801 words) - 20:34, 13 September 2016
The ''resultant of two polynomials $f(x)$ and $g(x)$''
is the element of the field $Q$ defined by the formula:
5 KB (859 words) - 11:30, 12 May 2022
...$ are variables and $A,B,\dots,D$ (the ''coefficients'' of the polynomial) and $k,l,\dots,t$ (the ''exponents of the powers'', which are non-negative inte
...same way it is possible to introduce or omit terms with zero coefficients and, in each individual term, zero powers. When the polynomial has one, two or
9 KB (1,497 words) - 10:44, 27 June 2015
...ment $w \in E$ that is simultaneously free in $E/K$ for every intermediate field $K$.
...ments of maximal multiplicative order, cf. [[Primitive element of a Galois field]]), see [[#References|[a4]]]
2 KB (296 words) - 19:12, 24 November 2023
...distinct primes, $\alpha_i \ge 0\,\,\ldots\,\,\nu_i \ge 0$, $i=1,\ldots,s$ and $\delta_i = \min\{\alpha_i,\ldots,\nu_i\}$, then
...actor, with a number of steps bounded by the smaller of the degrees of the polynomials: cf [[Euclidean ring]]
4 KB (673 words) - 17:01, 26 October 2014
Gröbner bases are certain sets of multivariate polynomials with field coefficients. The importance of Gröbner bases stems from the fact that
...uced by structurally easy algorithms to the construction of Gröbner bases; and
6 KB (1,000 words) - 17:05, 7 July 2014
...alois theory]]), and of stating the conditions which ensure the existence (and non-existence) of such an extension over $k$.
...ith the given Galois group. Such equations exist for the symmetric groups, and also for the alternating groups. I. Schur constructed equations for the alt
4 KB (586 words) - 04:07, 25 February 2022
$#C+1 = 79 : ~/encyclopedia/old_files/data/A012/A.0102800 Appell polynomials
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11 KB (1,595 words) - 17:13, 2 January 2021
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A statement obtained by K. Hensel [[#References|[1]]] in the creation of the theory of $ p $-
3 KB (386 words) - 22:10, 5 June 2020
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of discrete valuations on the field of fractions (cf. [[Fractions, ring of|Fractions, ring of]]) $ K $
3 KB (424 words) - 22:15, 5 June 2020
...mi-group]] with unit (i.e. [[Monoid]]) satisfying the [[cancellation law]] and in which any non-invertible element $a$ is decomposable into a product of i
a = b_1 \cdots b_k\ \ \text{and}\ \ a = c_1 \cdots c_l
1 KB (153 words) - 16:17, 21 December 2014
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...a unit element (cf. [[Associative rings and algebras]]) in which all right and left ideals are principal, i.e. have the form $ aR $
5 KB (880 words) - 19:00, 9 January 2024
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where the two initial polynomials are $D _ { 0 } ( x , a ) = 2$ and $D _ { 1 } ( x , a ) = x$. A second closed form is given by
15 KB (2,207 words) - 16:45, 1 July 2020
In the field of complex numbers a quadratic equation has two solutions, expressed by rad
When $b^2>4ac$ both solutions are real and distinct, when $b^2<4ac$, they are complex (complex-conjugate) numbers, whe
2 KB (384 words) - 07:34, 18 December 2014
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for the polynomials $\sigma ( z )$, $\omega ( z )$, $\operatorname { deg } \omega ( z ) < \oper
8 KB (1,155 words) - 18:48, 26 January 2024
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...case of the coefficients and roots being elements of an arbitrary [[Field|field]] may also be considered.
18 KB (2,778 words) - 16:09, 1 April 2020
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...L ( X )$ be the set of all linear operators with domains and ranges in $X$ and let
4 KB (576 words) - 10:55, 2 July 2020
...ative ring]] and let $(A,B)$ be a pair of matrices of sizes $(n \times n)$ and $(n \times m)$, respectively, with coefficients in $R$. The pole assignment
In both cases, state feedback (see [[Automatic control theory]]), $u \mapsto u + Fx$, changes the pair $(A,B)$ to $(A+BF,B)$.
5 KB (799 words) - 20:38, 12 November 2017
...le$, where $\Omega$ denotes the signature of $T$ (cf. [[Model theory|Model theory]]; [[Structure(2)|Structure]]). If $T$ is substructure complete, then it is
...admit a root in a given interval, cf. also [[Real closed field|Real closed field]]).
3 KB (435 words) - 09:40, 26 November 2016
...//www.encyclopediaofmath.org/legacyimages/h/h110/h110290/h11029022.png" />
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
...math.org/legacyimages/h/h110/h110290/h11029054.png" />-Galois extension if
and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofma
13 KB (1,801 words) - 19:17, 12 January 2018
...in [[#References|[a2]]]. Let $p$ denote the characteristic of the residue field $Kv$ if it is a positive prime; otherwise, set $p=1$. Hypothesis A requires
...$ has a solution in $Kv$. Requirement ii) implies that $Kv$ is a [[perfect field]]. Using [[Galois cohomology|Galois cohomology]], G. Whaples [[#References|
3 KB (523 words) - 19:54, 1 January 2015
...ariables with coefficients in some fixed field or ring. Let $K$ be a field and $F_3(x_0,\ldots,x_n)$ a cubic form with coefficients in $K$ (one calls it a
...f cubic forms over an [[algebraically closed field]] $K$ is reduced to the theory of cubic hypersurfaces (see [[#References|[1]]]).
3 KB (526 words) - 11:44, 12 October 2023
The theory of eliminating unknowns from systems of algebraic equations. More precisely
...$x_1,\dots,x_k$, one also consider the homogeneous problem in elimination theory (the inhomogeneous problem is trivial in this case): Find the projection on
8 KB (1,363 words) - 16:04, 13 January 2021
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consisting of the principal ideals. The divisor class group is Abelian and is usually denoted by $ C ( A) $.
5 KB (820 words) - 19:36, 5 June 2020
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are called the roots of the recurrence, and the $ n _ {i} $
6 KB (908 words) - 06:04, 12 July 2022
...es the existence of algebraic numbers of degree $n$. All rational numbers, and only such numbers, are algebraic numbers of the first degree. The number $i
...aic numbers; this means that the set of all algebraic numbers is a [[Field|field]]. A root of a polynomial with algebraic coefficients is an algebraic numbe
10 KB (1,645 words) - 17:08, 14 February 2020
...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
''factoring polynomials''
12 KB (1,739 words) - 13:11, 26 March 2023
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of Laurent polynomials in the variables $ t _ {i} $.
12 KB (1,694 words) - 06:42, 26 March 2023
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and $ w( x) $
3 KB (516 words) - 10:42, 19 February 2021
...Bernoulli numbers are the values of the [[Bernoulli polynomials|Bernoulli polynomials]] at $x=0$: $B_n=B_n(0)$; they also often serve as the coefficients of the
L. Euler in 1740 pointed out the connection between Bernoulli numbers and the values of the Riemann zeta-function $\zeta(s)$ for even $s=2m$:
4 KB (684 words) - 18:44, 5 October 2023
...th.org/legacyimages/m/m110/m110160/m1101601.png" /> of the ordered [[
Field|
field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org
...ctions (cf. [[Analytic function|Analytic function]]) for which there exist
polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
9 KB (1,282 words) - 17:26, 22 February 2013
...rentiation: differential rings, differential modules, differential fields, and differential algebraic varieties.
...of the main objects of differential algebra is the algebra of differential polynomials $ {\mathcal F} \{ Y _{1} \dots Y _{n} \} $ ,
30 KB (4,468 words) - 18:44, 17 December 2019
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and satisfies the relation usually referred to as the [[Leibniz rule]]
6 KB (945 words) - 17:32, 5 June 2020
...the theory of field extensions (cf. [[Extension of a field|Extension of a field]]). Let $ K $
be some extension of a field $ k $ .
6 KB (793 words) - 17:24, 17 December 2019
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and equality $ \mathop{\rm rk} L = \mathop{\rm dim} L $
3 KB (460 words) - 08:09, 6 June 2020
...tive algebraic groups [[#References|[4]]], [[#References|[5]]], and in the theory of formal groups [[#References|[6]]]. Let $ A $
which are added and multiplied in accordance with the following rules: $$
17 KB (2,502 words) - 17:25, 22 December 2019
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...stance, if $K \subseteq L$ is a field extension (cf. also [[Extension of a field]]), then $L$ is a Fatou extension of $K$.
5 KB (828 words) - 11:51, 24 December 2020
$#C+1 = 48 : ~/encyclopedia/old_files/data/R077/R.0707310 Random field, generalized,
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7 KB (970 words) - 08:09, 6 June 2020
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and endowing $ U ^ {*} $,
8 KB (1,190 words) - 08:16, 20 January 2024
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of modules and any homomorphism $ \beta : P \rightarrow C $
7 KB (1,080 words) - 08:08, 6 June 2020
...nd numerical; for example, in the (numerical) solution of linear equations and eigenvalue problems. A few well-known factorizations are listed below.
...th $m\ge n$ over $\C$. Then there exist a unitary $(m\times m)$-matrix $Q$ and a right-triangular $m\times n$-matrix $R$ such that $A=QR$. Here, a right-t
9 KB (1,294 words) - 22:47, 28 February 2015