...y studies the cohomology of a [[Galois group|Galois group]]. Let $M$ be an Abelian group, let $G(K/k)$ be the
defined for a non-Abelian group $M$. Namely,
17 KB (2,785 words) - 22:37, 23 December 2014
...of) Wedderburn's theorem (cf. [[Wedderburn–Artin theorem|Wedderburn–Artin theorem]]), that every finite-dimensional$C^*$-algebra is isomorphic to the direct
is the countable Abelian group of formal differences of equivalence classes of projections in matrix
12 KB (1,766 words) - 06:39, 26 March 2023
...References|[a11]]]). Eklof and Trlifaj attribute the inspiration for their theorem to a construction of [[#References|[a9]]]. D. Quillen [[#References|[a10]]]
...gn="top">[a11]</td> <td valign="top"> L. Salce, "Cotorsion theories for Abelian groups" , ''Symp. Math.'' , '''23''' , Amer. Math. Soc. (1979) pp. 11–3
6 KB (1,031 words) - 16:55, 1 July 2020
is an Abelian [[Semi-group|semi-group]] and $ E $
...great importance in dynamical systems theory, owing to the Smale–Birkhoff theorem: A discrete-time [[Dynamical system|dynamical system]] containing a [[Homoc
3 KB (392 words) - 08:13, 6 June 2020
of Abelian varieties $ T _ {\mathbf C} ^ {N} $(
of Abelian surfaces $ T _ {\mathbf C} ^ {2} $
13 KB (1,808 words) - 22:15, 5 June 2020
is exact; for an Abelian group $ G $,
is free (Stalling's theorem, see [[Homological dimension|Homological dimension]]). If $ G $
19 KB (2,870 words) - 09:48, 26 March 2023
...arbitrary dimension) of the classical [[Riemann–Roch theorem|Riemann–Roch theorem]] (see [[#References|[2]]]). After higher algebraic $ K $-
...ield theory]] in higher dimensions describes the Galois group of a maximal Abelian extension of rational function fields of arithmetic schemes of dimension $
8 KB (1,120 words) - 20:06, 31 October 2023
...w} K_{n - 2} \stackrel{\partial}{\longrightarrow} \cdots $ is a complex of abelian groups, and a continuous mapping of spaces induces homomorphisms of their r
...([[#References|[4]]]), the latter was extended so as to apply to arbitrary abelian categories with enough injective objects, and became applicable to arithmet
12 KB (1,885 words) - 23:48, 23 April 2017
...proof (cf. [[Schneider method|Schneider method]]) is based on the addition theorem for the exponential function $e ^ { z _ { 1 } + z _ { 2 } } = e ^ { z _ { 1
...elliptic integrals of the first or second kind [[#References|[a9]]] and of Abelian integrals [[#References|[a10]]], including the transcendence of the values
9 KB (1,244 words) - 19:58, 8 February 2024
consists of functions on a locally compact Abelian group and $ {\mathcal A} $
...includes the classical problems of harmonic synthesis on a locally compact Abelian group (see [[Harmonic analysis, abstract|Harmonic analysis, abstract]]), wh
9 KB (1,382 words) - 08:22, 6 June 2020
....png" />. In the general case this problem is solved by the Mittag-Leffler theorem: On every non-compact Riemann surface there exists a meromorphic function w
...xistence of a meromorphic function with a given divisor is given by Abel's theorem (see [[#References|[2]]]).
44 KB (5,974 words) - 22:47, 29 November 2014
...has strictly increasing complexity function (this is the '''Morse–Hedlund theorem'''), so $p_u(n) \ge n+1$.
...nd even $n \ge 2$ respectively. There is an analogue of the Morse–Hedlund theorem: if the complexity of $L$ satisfies $p_L(n) \le n$ for some $n$, then $p_L$
6 KB (991 words) - 07:57, 18 November 2023
...ings coincide, and one must substitute fields for simple rings in the last theorem above.
...he ring of bounded operators on a Hilbert space. A Baer ring is said to be Abelian if all its idempotents are central, and (Dedekind) finite if $ xy = 1 $
14 KB (2,228 words) - 08:10, 6 June 2020
Galois group of the extension $K/k$. An extension is called Abelian if
its Galois group is Abelian.
6 KB (1,030 words) - 17:29, 12 November 2023
...homotopy classes of morphisms $B \rightarrow C$. A homotopy classification theorem is that if $X_{*}$ is the skeletal filtration of a [[CW-complex|CW-complex]
...rossed module|crossed module]] case. It also implies the relative Hurewicz theorem (an advantage of this deduction is its generalization to the $n$-adic situa
13 KB (1,937 words) - 13:10, 24 December 2020
is satisfied (cf. [[Amitsur–Levitzki theorem]]). A tensor product of PI-algebras is a PI-algebra.
have an Abelian subgroup of finite index. If the characteristic of $ F $
15 KB (2,252 words) - 08:04, 6 June 2020
...and W. Sikonia [[#References|[a12]]], working independently. However, the theorem is not true for operators that are only essentially normal, in other words,
...} )$) to $\mathcal{Q} ( \mathcal{H} )$, and the $C ^ { * }$-algebra $A$ is Abelian if and only if $T$ is essentially normal. More generally, an extension of a
10 KB (1,450 words) - 17:44, 1 July 2020
...theorem [[#References|[a20]]] (cf. also [[Fermat last theorem|Fermat last theorem]]). For details and generalizations of Iwasawa theory, see [[#References|[a
...ery prime ideal $\mathfrak{p}$ of $k$ lying above $p$. By Dirichlet's unit theorem, $\operatorname{rank}_{\mathbf{Z}} E _ { 1 } ( k ) = r _ { 1 } ( k ) + r _
19 KB (2,876 words) - 05:38, 15 February 2024
...categories one considers additive functors with values in the category of Abelian groups instead of functors with values in $ \mathfrak S $.
is an Abelian variety over $ K $,
6 KB (837 words) - 07:19, 14 November 2023
...athbb{N}}(n)$ for large $x$ forms the content of the famous [[prime number theorem]], which states that
...eorem|de la Vallée-Poussin theorem]]). A suitably generalized form of this theorem holds for many other naturally-occurring arithmetical semi-groups. For exam
24 KB (3,738 words) - 07:41, 7 February 2024