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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
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''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