...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]]
345 bytes (66 words) - 12:43, 14 February 2020
''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