...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]].
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...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 $
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...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
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...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
...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 $
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...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 ^ {*} $,
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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
''Associative rings and algebras'' are
rings and algebras with an associative multiplication, i.e., sets with
11 KB (1,726 words) - 20:09, 15 December 2020
and the solution to this equation may be obtained by using
Ferro's result; he also solved the equation $x^3+px+q$ ($(p>0$, $q>0$), and
2 KB (312 words) - 19:38, 15 March 2023
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...ce in which the modulus is a prime number. A distinguishing feature of the theory of congruences modulo a prime number is the fact that the residue classes m
7 KB (1,033 words) - 17:46, 4 June 2020
...articular, in a ring of polynomials in a finite number of variables over a field or over a ring of integers any ideal is generated by a finite number of ele
be an irreducible polynomial over the field $ \mathbf Q $
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...among the widely available systems is REDUCE, which runs on many platforms and is still being further developed.
...1983. Maple was released in 1986, while Scratchpad II (later called AXIOM) and Mathematica appeared by 1988.
9 KB (1,292 words) - 20:51, 18 September 2016
a linear algebraic group over some field, related to a semi-simple
Chevalley basis of the algebra $\fg$, and let $\fg_\Z$ be its linear
6 KB (1,002 words) - 19:25, 3 November 2013
A [[Differential field|differential field]] $ F \supset F _{0} $
and $ F _{0} $;
13 KB (1,981 words) - 20:02, 17 December 2019
...ring, there exists a one-to-one correspondence between varieties of rings and the $T$-ideals of a countably-generated free ring. If for two varieties of
...bra with a finite number of generators over a field of characteristic zero and if $M_2\not\subseteq\mathfrak M$, then $\mathfrak M$ is a Specht variety.
4 KB (701 words) - 17:01, 23 November 2023
$#C+1 = 70 : ~/encyclopedia/old_files/data/D033/D.0303860 Double and dual numbers
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6 KB (895 words) - 19:36, 5 June 2020
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of polynomials over a field $ F $,
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...{ q }$ which contains $V$ (cf. also [[Extension of a field|Extension of a field]]). Then the monic separable polynomial whose roots are precisely the eleme
6 KB (860 words) - 16:28, 12 July 2020
...lowing more general assertion, which was stated without proof by Lindemann and was proved by K. Weierstrass in 1885, is known as the Lindemann–Weierstra
Let $a_1,\dots,a_m$ be non-zero algebraic numbers and let $\alpha_1,\dots,\alpha_m$ be pairwise distinct algebraic numbers; then
3 KB (379 words) - 15:19, 19 August 2014
...$f_1, \ldots, f_k$, then the curve $L$ defined by (1) is called reducible and is the union of the curves $L_1, \ldots, L_k$ (the components of $L$) defin
...cible curve. Two irreducible plane real algebraic curves, one of order $n$ and the other of order $m$, intersect in at most $mn$ points (Bezout's theorem)
11 KB (1,916 words) - 00:44, 12 August 2019
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and of a rank lower than $ t $,
4 KB (563 words) - 17:28, 26 December 2020
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...{ 0 }$ is a [[Polynomial|polynomial]] in $X$ with coefficients in $\bf Z$ and there exists an $x \in \widetilde{\mathbf{Z}}$ such that $f ( x )$ is a uni
11 KB (1,771 words) - 16:57, 1 July 2020
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be a field which is complete with respect to an [[ultrametric]] valuation $ | \cdot
8 KB (1,161 words) - 08:25, 6 June 2020
...f Lie algebras, i.e., $ \sigma $ is a $ \mathbb{k} $-linear transformation and $ \sigma([x,y]) = \sigma(x) \sigma(y) - \sigma(y) \sigma(x) $ for all $ x,y
# For every associative $ \mathbb{k} $-algebra $ A $ with a unit element and every $ \mathbb{k} $-algebra homomorphism $ \alpha: \mathfrak{g} \to A $ su
6 KB (970 words) - 18:59, 5 April 2023
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...structure. While the theory of analytic surfaces forms part of the general theory of complex manifolds, the two-dimensional case is treated separately, since
8 KB (1,169 words) - 18:47, 5 April 2020
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...o account both the geometry of the domain on which the function is defined and the behaviour of the function while approaching the boundary, so that the s
21 KB (3,060 words) - 20:29, 10 January 2021
...dges, which may be replaced by a triangle (A. Adler, 1890); a straightedge and a circle with a marked centre given in the plane of the figure (Poncelet–
..., $k \in \mathbf{N}$. For example, the regular $n$-gon is constructible if and only if each odd prime power dividing $n$ is a [[Fermat prime]] $2^{2^m} +
5 KB (738 words) - 19:10, 17 December 2015
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...fference set|Difference set]] (Vol. 3). For an extensive discussion of the theory of Abelian difference sets, see also [[#References|[a1]]], Chap. VI.
9 KB (1,331 words) - 19:36, 13 February 2024
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...\mathbf{C} \backslash \{ 0 , \infty \}$ which have algebraic poles at $0$ and $\infty$. A basis of the algebra is given by the set
9 KB (1,398 words) - 10:21, 11 November 2023
[[Category:Linear and multilinear algebra; matrix theory]]
consisting of $m$ rows and $n$ columns, the
18 KB (3,377 words) - 17:54, 2 November 2013
In the most general sense, ''Galois theory''
is a theory dealing with mathematical
11 KB (1,965 words) - 04:47, 16 January 2022
see [[Algebraic K-theory|Algebraic $ K $-
theory]]).
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In the simplest and most often used case, a [[Hilbert space|Hilbert space]] consisting of infin
11 KB (1,537 words) - 19:39, 5 June 2020
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...concept of an arithmetical semi-group $G$ (cf. [[Abstract analytic number theory]]; [[Semi-group]]).
14 KB (2,037 words) - 09:45, 11 November 2023
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...domains where the [[Ray method|ray method]] cannot be employed because the field of rays suffers from a singularity in one sense or another. Consider, for e
8 KB (1,156 words) - 08:04, 6 June 2020
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and $ X $ (cf. [[Chern class|Chern class]]). It can be used to calculate the
10 KB (1,385 words) - 03:10, 2 March 2022
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with coefficients in some field $ K $,
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...p$, the appropriate generalization is not just to take $x$, $y$ and $z$ as polynomials over $\mathbf{F}$ in $T$. In any event, in characteristic zero, or for $n$
4 KB (702 words) - 16:45, 1 July 2020
...nnected with solving inequalities in integers — Diophantine inequalities — and also with solving equations in integers (cf. [[Diophantine equations|Diopha
...d03260013.png" />-adic number fields and Diophantine approximations in the field of power series).
54 KB (7,359 words) - 18:32, 31 March 2017
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...l tool in many applications, particularly communication science and coding theory.
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with identity over the field $ \mathbf C $
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...lead to abstract generalizations and analogues of ordinary analytic number theory, which may then be applied in a unified way to further enumeration question
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''Galois field (update)''
14 KB (2,032 words) - 14:35, 19 March 2023
...obabilistically checkable" is somewhat misleading, since both holographic and traditional proofs can be checked either deterministically or probabilistic
...oice. This chance never exceeds $50$%, regardless of the nature of errors, and vanishes as $1/2^k$ with $k$ independent rounds.
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''(in the classical sense), invariant theory''
...dies on the theory of binary quadratic forms posed the problem of studying polynomials in the coefficients of the form $ ax ^{2} + 2 b xy + c y ^{2} $
22 KB (3,406 words) - 07:08, 6 May 2022
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The field concerned with best rational approximation to power series. Let
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In [[#References|[a2]]], M. Eichler conceived the "Eichler cohomology theory" (but not the designation) while studying "generalized Abelian integrals"
13 KB (1,993 words) - 07:12, 15 February 2024
where $i_1,\dots,i_n$ is a permutation of the numbers $1,\dots,n$ and $k$ is the number of inversions of the permutation $1\mapsto i_1,\dots,n\ma
...eld (especially a number field), a ring of functions (especially a ring of polynomials) or a ring of integers.
11 KB (1,876 words) - 20:27, 30 November 2016
{{Cite|Ch}} on conics and was systematized and used to great effect by H. Schubert in
{{Cite|Sc}}. The justification of Schubert's enumerative calculus and the verification of the numbers he obtained was the contents of Hilbert's 1
8 KB (1,263 words) - 08:49, 30 March 2012
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The branch of algebra studying the properties of commutative rings and objects relating to them (ideals, modules, valuations, etc., cf. [[Ideal|Id
16 KB (2,400 words) - 17:45, 4 June 2020
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and $ m $
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and $n$ are natural numbers; the $a_{ij}$ ($i=1,\dots,m$; $j=1,\dots,n$) are ca
coefficients at the unknowns and are given; the $b_i$ ($i=1,\dots,m$) are called
12 KB (2,085 words) - 22:05, 5 March 2012
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An algebra over a field for which certain polynomial identities are true.
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The geometry of the Yang–Mills equations (cf. [[Yang–Mills field|Yang–Mills field]]) led to deep purely mathematical insight, some of which is given below. F
7 KB (1,126 words) - 17:45, 1 July 2020
and the structure of an associative graded [[Co-algebra|co-algebra]] $ \delta
and $ B $
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of this sequence and only at these points. This function may be constructed as a [[Canonical pro
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...4]]] on the other. Another approach has been undertaken by Yu. Berezanskii and collaborators [[#References|[a1]]].
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