component of the identity of the Abelian
Abelian variety, the concept of the degree of polarization of a
4 KB (644 words) - 13:06, 17 April 2023
with semi-integer characteristics one can construct meromorphic Abelian functions with $ 2p $
periods. The periods of an arbitrary Abelian function in $ p $
14 KB (1,941 words) - 05:01, 23 February 2022
i) The uniform convergence theorem: for $ f $
ii) The representation theorem: $ f $
6 KB (794 words) - 22:14, 5 June 2020
of étale Abelian sheaves $ F _ {n} $
is an [[Abelian scheme|Abelian scheme]] over $ X $,
6 KB (932 words) - 11:49, 8 April 2023
...Minkowski addition leads to the [[Brunn–Minkowski theorem|Brunn–Minkowski theorem]] and is the basis for the Brunn–Minkowski theory of convex bodies (i.e.,
...s a convexifying effect; this is made precise by the Shapley–Folkman–Starr theorem.
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cobordism theorem [[#References|[4]]]). Thus, proving the isomorphism $ M _ {0} \approx M _
...ch can be achieved by methods of algebraic topology. For this reason, this theorem is basic in passing from the homotopy classification of simply-connected ma
10 KB (1,458 words) - 07:41, 10 February 2024
By the Dirichlet unit theorem (cf. also [[Dirichlet theorem|Dirichlet theorem]]), the unit group $ U _ {F} $
that are Abelian over $ \mathbf Q $
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...y other class of fields. For imaginary quadratic fields, the Brauer–Siegel theorem (stating that for algebraic number fields of fixed degree the following asy
...ory of complex multiplication (see {{Cite|CaFr}}) enables one to construct Abelian extensions of imaginary quadratic fields in an explicit form.
5 KB (867 words) - 17:41, 12 November 2023
...x)$ is divisible (without remainder) by $x-c$ (see [[Bezout theorem|Bezout theorem]]). Every polynomial $f(x)$ with real or complex coefficients has at least
...a field $k$ are roots of unity (cf. [[Fermat little theorem|Fermat little theorem]]) and the subgroup itself is cyclic. This is true, in particular, for the
4 KB (680 words) - 13:40, 30 December 2018
...nd has a complement (see [[Krull–Remak–Schmidt theorem|Krull–Remak–Schmidt theorem]]).
...$ is a perfect ring and $G$ is a finite group. The endomorphism ring of an Abelian group $A$ is perfect only when $A$ is the direct sum of a finite group and
3 KB (491 words) - 19:59, 30 October 2016
An Abelian algebra is nilpotent. If $ V $
is Abelian. The unique non-Abelian three-dimensional nilpotent Lie algebra $ \mathfrak g $
10 KB (1,457 words) - 18:48, 13 January 2024
It is clear from Stokes' theorem (cf. [[Stokes theorem|Stokes theorem]]) that the integral $ \int _ {C} A $
extends to the non-Abelian or Yang–Mills case.
8 KB (1,229 words) - 08:30, 26 March 2023
...of the group $ G $ of automorphisms which preserve the structure of the Abelian variety, by the group $ A(k) $ of translations in the points of $ A $
...Grothendieck gave a proof of this fact for projective varieties, and this theorem has been extended to the case of proper flat schemes of morphisms. The sche
6 KB (923 words) - 18:31, 12 December 2019
...tryagin [[#References|[2]]] on the theory of characters of locally compact Abelian groups (cf. [[Character of a group|Character of a group]]), posed the probl
...veloped mainly on the basis of the theory of characters of locally compact Abelian groups established by Pontryagin ([[#References|[2]]], see also [[#Referenc
66 KB (9,085 words) - 17:28, 31 March 2020
Using this theorem, it can be proved that every $ l $-
group is large. E.g., it contains the classes of Abelian torsion-free groups, locally nilpotent torsion-free groups, and many others
9 KB (1,403 words) - 22:15, 5 June 2020
$#C+1 = 45 : ~/encyclopedia/old_files/data/P072/P.0702770 Plancherel theorem
denotes the inverse, then Plancherel's theorem can be rephrased as follows: $ {\mathcal F} $
5 KB (752 words) - 13:19, 20 March 2023
...f bad reduction (see [[#References|[4]]], and also [[Siegel theorem|Siegel theorem]] on integer points).
...famous conjectures, namely the Tate conjecture concerning endomorphisms of Abelian varieties over number fields (cf. [[Tate conjectures|Tate conjectures]]) an
7 KB (1,068 words) - 08:01, 6 June 2020
...e group. Every subgroup of a free group is also free (the Nielsen–Schreier theorem, see [[#References|[1]]], [[#References|[2]]]).
Free groups of certain varieties have special names, for example, free Abelian, free nilpotent, free solvable, free Burnside; they are free groups of the
4 KB (572 words) - 19:40, 5 June 2020
...elian group]] are of the form $\pm g$, $g \in G$. Of course, if $G$ is non-Abelian, then any conjugate of $\pm g$ is also of finite order; however, these are
...r, it was proved in [[#References|[a6]]] that if $n = 2$ and $G$ is finite Abelian, then $U$ is conjugate in $\mathbf{Q}G_{2\times 2}$ to $\operatorname{diag}
9 KB (1,457 words) - 17:05, 26 January 2021
$#C+1 = 29 : ~/encyclopedia/old_files/data/P071/P.0701120 Paley\ANDWiener theorem
...ner theorem; the most frequently encountered analogues of the Paley–Wiener theorem are a description of the image of the space $ C _ {0} ^ \infty ( G) $
4 KB (591 words) - 08:04, 6 June 2020