...\mathbf{Q}$ such that the [[Galois group]] $\mathrm{Gal}(K/\mathbf{Q})$ is Abelian. Examples include: the quadratic number fields $\mathbf{Q}(\sqrt{d})$ and
...such that $K$ is contained in $\mathbf{Q}(\zeta_n)$, cf. [[Conductor of an Abelian extension]].
813 bytes (123 words) - 20:47, 23 November 2023
..., then the validity of P in some particular category (e.g. the category of
Abelian groups or the category of sets) implies its validity in all categories of t
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Freyd, "
Abelian categories: An introduction to the theory of functors" , Harper & Row
1 KB (173 words) - 17:22, 7 February 2011
An [[Abelian group|Abelian group]] is cotorsion-free if it does not contain any non-zero [[Cotorsion g
...cyclic group]] of prime order (thus, it is torsion-free). Equivalently, an Abelian group $ A $
4 KB (561 words) - 17:31, 5 June 2020
...bgroup series]]). It also possesses a [[Normal series|normal series]] with Abelian quotient groups (such series are called solvable). The length of the shorte
...er. These groups are characterized by the following converse to Lagrange's theorem: For any factorization $n=n_1n_2$ of the order $n$ of a group into two rela
3 KB (443 words) - 18:25, 26 October 2014
...idèle class group $C_K$ (cf. [[Class field theory]]). The conductor of an Abelian extension is the greatest common divisor of all positive divisors $n$ such
...extension is given by the theorem that the conductor $\mathfrak{f}$ of an Abelian extension $L/K$ of number fields is equal to $\prod_{\mathfrak{p}} \mathfra
2 KB (440 words) - 19:42, 7 March 2018
An [[Abelian group|Abelian group]] $ C $
for all [[Torsion-free group|torsion-free]] Abelian groups $ G $,
4 KB (674 words) - 17:31, 5 June 2020
has an elementary Abelian subgroup of order $ p ^ {n} $(
cf. [[Abelian group|Abelian group]]). A $ p $-
4 KB (568 words) - 14:10, 31 December 2020
[[Brauer first main theorem|Brauer first main theorem]].
...gers with $\def\a{\alpha}\nu(np^\a)=\a$ whenever $n$ is prime to $p$. By a theorem of Brauer, $\nu(\chi(1)\ge \nu(|G:D|)$. The height of $\chi$ is defined to
2 KB (352 words) - 14:21, 13 April 2012
...categories similar to an [[Abelian category|Abelian category]]. In the non-Abelian case the direct sum is usually called the discrete direct product. Let $
group (in particular, for groups, Abelian groups, vector spaces, and rings) one can give an "intrinsic" characteriz
4 KB (680 words) - 19:35, 5 June 2020
For finite-dimensional Lie algebras over a field of characteristic 0, Lévy's theorem holds: If $ S $
...ions, the Abelian ones have been studied most, i.e. the extensions with an Abelian kernel $ A $.
3 KB (416 words) - 12:53, 19 March 2023
$#C+1 = 60 : ~/encyclopedia/old_files/data/A110/A.1100020 Abelian difference set
...n (cyclic, non-Abelian), the difference set is called Abelian (cyclic, non-Abelian). Two difference sets $ D _ {1} $
5 KB (803 words) - 06:20, 26 March 2023
...ecreased. The Thue–Siegel–Roth theorem is a strengthening of the Liouville theorem (see [[Liouville number|Liouville number]]). Liouville's result has been su
...was obtained by Roth. There is a $p$-adic analogue of the Thue–Siegel–Roth theorem. The results listed above are proved by non-effective methods (see [[Diopha
4 KB (557 words) - 18:10, 23 November 2014
The [[characteristic subgroup]] of a [[P-group|$p$-group]] generated by all Abelian subgroups of maximal order. Introduced by J.G. Thompson [[#References|[1]]]
...D valign="top">[1]</TD> <TD valign="top"> J.G. Thompson, "A replacement theorem for $p$-groups and a conjecture" ''J. Algebra'' , '''13''' (1969) pp. 14
724 bytes (97 words) - 14:51, 8 April 2023
...Abelian group. Examples of commutative group schemes are [[Abelian scheme|Abelian schemes]] and [[Algebraic torus|algebraic tori]]. A generalization of algeb
$$ where $M$ is an Abelian group and ${\mathcal O}_S(M)$ is its group algebra with coefficients in the
4 KB (629 words) - 20:08, 15 December 2020
...ory of Abelian groups and the category that is dual to that of topological Abelian groups; the category of Boolean algebras is equivalent to the category that
1 KB (231 words) - 07:37, 28 November 2017
$#C+1 = 92 : ~/encyclopedia/old_files/data/A010/A.0100240 Abelian integral,
of the [[Abelian differential|Abelian differential]] $ \omega = R (z, w) dz $
10 KB (1,594 words) - 06:20, 17 April 2024
...ferences|[a2]]], M. Hall proved the following generalization of Frobenius' theorem: If $G$ is a finite group of order $g$ and $C$ is a [[conjugacy class]] of
...ns. Thus, the conjecture holds in Abelian groups (cf. also [[Abelian group|Abelian group]]). It is also easy to see that it suffices to show that $G$ contains
4 KB (650 words) - 20:59, 29 November 2014
On the other hand, there exists an amalgam of four Abelian groups that is not imbeddable in a group. The principal problem concerning
...algam of five Abelian groups which is imbeddable in a group, but not in an Abelian group. Another problem that has been studied is the imbeddability of an ama
5 KB (883 words) - 16:10, 1 April 2020
...uations by means of radicals. It is customary to write the operation in an Abelian group in additive notation, i.e. to use the plus sign ($+$) for that operat
...up]]s $\mathbf Z_{p^\infty}$), where $p$ is an arbitrary prime number, are Abelian (cf. [[Group-of-type-p^infinity|Group of type $p^\infty$]]).
11 KB (1,810 words) - 22:12, 29 August 2015
...clopedia/old_files/data/G110/G.1100050 Gamma\AAhinvariant in the theory of Abelian groups,
...ubgroup of strictly smaller cardinality is a [[free Abelian group]]). By a theorem of S. Shelah (see [[#References|[a7]]]), such a group is free if it is of [
6 KB (845 words) - 19:41, 5 June 2020
$#C+1 = 90 : ~/encyclopedia/old_files/data/A010/A.0100200 Abelian category
...me of the characteristic properties of the category of all Abelian groups. Abelian categories were introduced as the basis for an abstract construction of hom
10 KB (1,515 words) - 18:19, 31 March 2020
...blem attributed, to J.H.C. Whitehead, which asks for a characterization of Abelian groups $ A $
is free (see [[Free Abelian group|Free Abelian group]]). This condition has been proved to be necessary if $ A $
4 KB (665 words) - 08:29, 6 June 2020
$#C+1 = 42 : ~/encyclopedia/old_files/data/F041/F.0401790 Frobenius theorem
...ors of zero; it was proved by G. Frobenius [[#References|[1]]]. Frobenius' theorem asserts that:
5 KB (790 words) - 19:40, 5 June 2020
$#C+1 = 145 : ~/encyclopedia/old_files/data/A010/A.0100210 Abelian differential
...g on the nature of their singular points, one distinguishes three kinds of Abelian differentials: I, II and III, with proper inclusions $ I \subset II \sub
11 KB (1,603 words) - 16:08, 1 April 2020
Then the category of sheaves of Abelian groups on $ X _ {et} $
is an Abelian category with a sufficient collection of injective objects. The functor $
5 KB (746 words) - 11:54, 8 April 2023
of an [[Abelian category|Abelian category]] $ \mathfrak A $
is Abelian.
3 KB (469 words) - 16:39, 17 March 2023
extensions with an Abelian Galois group (Abelian extensions) is a part
The fundamental result on Galois groups is the following theorem,
3 KB (494 words) - 21:56, 5 March 2012
...are isomorphic over a finite extension of $K$. One of the marvels of this theorem is the fact that the construction of the period $q$ starting from $E$, and
...], and it was used in the theory of compactifications of moduli schemes of Abelian varieties.
4 KB (680 words) - 21:50, 21 December 2014
$#C+1 = 99 : ~/encyclopedia/old_files/data/A010/A.0100220 Abelian function
is called an Abelian function if there exist $ 2p $
11 KB (1,602 words) - 16:08, 1 April 2020
coincides with the set of one-dimensional non-Abelian cohomology $ H ^{1} ( S _{T} ,\ \Gamma ) $.
group is trivial (Lang's theorem). This theorem also holds if $ k $
5 KB (854 words) - 10:51, 20 December 2019
in an [[Abelian category|Abelian category]] $ C $
1) The category of Abelian groups has enough injective objects. These objects are the complete (divisi
4 KB (643 words) - 22:12, 5 June 2020
The classical Torelli theorem relates to the case of curves (see [[#References|[1]]], [[#References|[2]]]
be a basis of the Abelian differentials (cf. [[Abelian differential|Abelian differential]]) and let the $ ( g \times 2g) $-
6 KB (967 words) - 08:26, 6 June 2020
A theorem proved by J.-P. Serre in 1965 about the cohomology of pro-$p$-groups which
...dexing set $I$, where $\textbf{Z}/p$ is cyclic of order $p$). Then Serre's theorem asserts that there exist non-trivial $\mod p$ cohomology classes $v_1,...,v
6 KB (868 words) - 22:16, 5 February 2021
...iaofmath.org/legacyimages/f/f120/f120130/f12013073.png" /> with elementary
Abelian quotient groups <img align="absmiddle" border="0" src="https://www.encyclop
...opediaofmath.org/legacyimages/f/f120/f120130/f12013092.png" />, then, by a
theorem of Burnside, <img align="absmiddle" border="0" src="https://www.encyclopedi
16 KB (2,143 words) - 17:10, 7 February 2011
...larized algebraic variety|Polarized algebraic variety]]; [[Abelian variety|Abelian variety]]), which is not always true for $ T _ {G} ^ {n} ( X) $.
as well as a duality between the Abelian varieties $ T _ {W} ^ {n} ( X) $
6 KB (953 words) - 12:29, 29 December 2021
...of Orlicz). The result subsequently came to be known as the Orlicz–Pettis theorem (see [[#References|[a3]]] for a historical discussion).
...measure and integration theory, there have been attempts to generalize the theorem in several directions. For example, A. Grothendieck remarked that the resul
5 KB (714 words) - 15:30, 1 July 2020
from an (Abelian) [[Semi-group|semi-group]] $ H $
to subsets of an (Abelian) semi-group $ G $
2 KB (318 words) - 16:09, 1 April 2020
be an [[Abelian group|Abelian group]] and let $ A \subset G $.
...degree of the minimal polynomial of the Grasmann derivative, the following theorem is true [[#References|[a3]]]: Let $ p $
4 KB (577 words) - 10:26, 10 December 2023
...nsion $K/k$ is Kummer (for a given $n$) if and only if $K/k$ is a normal [[Abelian extension]] and the Galois group $\mathrm{Gal}(K/k)$ is annihilated by $n$.
...$\mathrm{Gal}(K/k_0)$.) By the above proposition, many problems concerning Abelian extensions of exponent $n$ of a field $k$ can be reduced to the theory of K
5 KB (938 words) - 20:00, 18 September 2017
...ance of loops in the theory of quasi-groups is determined by the following theorem: Any quasi-group is isotopic (see [[Isotopy|Isotopy]]) to a loop. Therefore
Albert's theorem). In particular, isotopic groups are isomorphic. Some other classes of loop
8 KB (1,291 words) - 06:59, 30 March 2024
An Abelian variety is
implies severe restrictions on an Abelian variety. Thus, an Abelian
8 KB (1,216 words) - 20:39, 5 March 2012
$#C+1 = 17 : ~/encyclopedia/old_files/data/C023/C.0203650 Comparison theorem (algebraic geometry)
A theorem on the relations between homotopy invariants of schemes of finite type over
2 KB (271 words) - 13:16, 6 April 2023
Khinchin's theorem on the factorization of distributions: Any probability distribution $P$ adm
...tions on the line, in which factorization theorems analogous to Khinchin's theorem are valid.
2 KB (326 words) - 16:26, 9 April 2016
...is denoted by $\mathrm{NS}(X)$. The Néron–Severi theorem asserts that the Abelian group $\mathrm{NS}(X)$ is finitely generated.
...heory of the base (see, for example, [[#References|[1]]]), a proof of this theorem using topological and transcendental tools. The first abstract proof (valid
4 KB (687 words) - 05:47, 15 April 2023
...additive category]] with set of objects $\mathrm{Ob}(C)$ and let $G$ be an Abelian group. A mapping $\phi: \mathrm{Ob}(C) \to G$ is said to be additive if for
...f coherent and locally free sheaves on schemes in proving the Riemann–Roch theorem. See [[K-functor|$K$-functor]] in algebraic geometry. The group $K(C)$ is u
4 KB (701 words) - 06:11, 26 March 2023
''EGZ theorem''
...\{1,\ldots,2m-1\}$ of cardinality $m$ such that $\sum_{i\in I}a_i=0$. This theorem was first shown in [[#References|[a5]]].
10 KB (1,573 words) - 17:25, 28 January 2020
consisting of the principal ideals. The divisor class group is Abelian and is usually denoted by $ C ( A) $.
Nagata's theorem). If $ B $
5 KB (820 words) - 19:36, 5 June 2020
$#C+1 = 187 : ~/encyclopedia/old_files/data/O110/O.1100050 O\AApNan\ANDScott theorem
A reduction theorem for the class of finite primitive permutation groups, distributing them in
11 KB (1,611 words) - 08:03, 6 June 2020
...s a description of all Abelian extensions (finite Galois extensions having Abelian Galois groups) of a field $ K $
In local class field theory, each finite Abelian extension $ L/K $
17 KB (2,620 words) - 07:48, 13 February 2024
...2 : ~/encyclopedia/old_files/data/K055/K.0505920 Krull\ANDRemak\ANDSchmidt theorem
...or a ring. The lattice-theoretical version of the result is known as Ore's theorem (see [[Modular lattice|Modular lattice]]). For a group $ G $
5 KB (786 words) - 22:15, 5 June 2020
There are no entire elliptic functions except the constants (Liouville's theorem).
are integers (a special case of Abel's theorem, see [[Abelian function|Abelian function]]).
9 KB (1,292 words) - 19:08, 20 January 2022
...ediaofmath.org/legacyimages/b/b015/b015310/b01531041.png" /> is a sheaf of
Abelian groups, then for every <img align="absmiddle" border="0" src="https://www.e
...hemes — see [[#References|[2]]] — can also be interpreted as a base-change
theorem); or 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofma
11 KB (1,513 words) - 17:08, 7 February 2011
[[Abelian variety|Abelian variety]] (cf. also
[[Abelian differential|Abelian differential]]). The polarization class
10 KB (1,582 words) - 22:02, 5 March 2012
...belian varieties and behaves as a functor with respect to the morphisms of Abelian varieties preserving the zero point. For the local aspect see [[#References
...ess (1974) (Appendix in Russian translation: Yu.I. Manin; The Mordell–Weil theorem (in Russian)) {{MR|2514037}} {{MR|1083353}} {{MR|0352106}} {{MR|0441983}} {
6 KB (893 words) - 08:43, 1 May 2023
...gory|Topologized category]]) in dimensions 0, 1. A unified approach to non-Abelian cohomology can be based on the following concept. Let $ C ^{0} $,
fixed. Then a non-Abelian cochain complex is a collection $$
12 KB (1,712 words) - 09:30, 20 December 2019
...maximum condition for subgroups is equivalent to the maximum condition for Abelian subgroups [[#References|[4]]]. A similar result was also established for th
...ups and locally nilpotent groups. It was found, in particular, that if all Abelian subgroups of a locally nilpotent torsion-free group (cf. [[Group without to
5 KB (720 words) - 17:15, 7 February 2011
...y existence condition. A non-trivial restriction is the Bruck–Ryser–Chowla theorem, see [[Block design|Block design]]. This condition is not sufficient, as th
...method for symmetric designs which combines Abelian difference sets (cf. [[Abelian difference set]]; [[Difference set|Difference set]]; [[Difference set|Diffe
7 KB (1,152 words) - 16:45, 1 July 2020
A nilpotent, in particular an Abelian, Lie group is solvable. If $ F = \{ V _{i} \} $ is a complete [[Flag|fla
An analogue of Lie's theorem on solvable Lie algebras is true for solvable Lie groups: If $ \rho : \
7 KB (1,043 words) - 18:17, 12 December 2019
...between [[topological group]]s and their [[character group]]s. The duality theorem states that if $ G $
is a locally compact Abelian group and if $ X ( G) $
10 KB (1,483 words) - 17:06, 13 June 2020
$#C+1 = 70 : ~/encyclopedia/old_files/data/L058/L.0508000 Lefschetz theorem
Lefschetz' fixed-point theorem, or the Lefschetz–Hopf theorem, is a theorem that makes it possible to express the number of fixed points of a continuou
11 KB (1,584 words) - 11:51, 8 April 2023
$#C+1 = 261 : ~/encyclopedia/old_files/data/A110/A.1100040 Abelian surface
An [[Abelian variety|Abelian variety]] of dimension two, i.e. a complete connected group variety of dime
18 KB (2,511 words) - 06:25, 26 March 2023
and an Abelian group $ G $.
see [[De Rham theorem|de Rham theorem]]).
16 KB (2,386 words) - 16:47, 20 January 2024
...lpotent as an abstract group (cf. [[Nilpotent group|Nilpotent group]]). An Abelian Lie group is nilpotent. If $ F = \{ V _{i} \} $ is a [[Flag|flag]] in a
...p version of Engel's theorem admits the following strengthening (Kolchin's theorem): If $ G $ is a subgroup of $ \mathop{\rm GL}\nolimits (V) $ , where
5 KB (803 words) - 18:12, 12 December 2019
The Mittag-Leffler theorem on expansion of a meromorphic function (see , ) is one of the basic theorem
The Mittag-Leffler theorem implies that any given meromorphic function $f(z)$ in $\mathbb{C}$ with pol
6 KB (980 words) - 18:47, 24 May 2017
...also [[Graph automorphism|Graph automorphism]]). This is know as Frucht's theorem. In 1949, Frucht [[#References|[a3]]] extended this result by showing that
...ese results see [[#References|[a4]]]. For an alternative proof of Frucht's theorem see [[#References|[a5]]].
3 KB (387 words) - 09:29, 19 January 2021
...oduct|Direct product]]) of $R$-groups, are $R$-groups. The following local theorem is valid for the class of $R$-groups: If all finitely-generated subgroups o
1 KB (254 words) - 11:47, 29 June 2014
Theorem 1) was later strengthened; namely, it was proved that the condition $ a _
...should mean. In [[#References|[a1]]], p. 195, is written: " a theorem is Abelian if it says something about an average of a sequence from a hypothesis about
11 KB (1,603 words) - 10:19, 7 May 2021
...athbf{Z}$, then this is true only under the condition that $L/M$ is a free Abelian group [[#References|[2]]]. The finitely-generated subalgebras of a free Lie
...belong to $L(X)$ are given by the Specht–Wever theorem and the Friedrichs theorem, respectively. The first one says that a homogeneous element $a$ of degree
3 KB (564 words) - 19:53, 15 March 2023
...uction associated with special radical subcategories; it first appeared in Abelian categories in the description of the so-called Grothendieck categories in t
be an [[Abelian category|Abelian category]]. A full subcategory $ {\mathfrak A ^ \prime } $
10 KB (1,375 words) - 22:17, 5 June 2020
...n a number of areas of analysis. If $\{x_k\}$ is a sequence in a Hausdorff Abelian [[Topological group|topological group]] $(G,\tau)$, then $\{x_k\}$ is $\tau
...rem|Banach–Steinhaus theorem]] and the [[Mazur–Orlicz theorem|Mazur–Orlicz theorem]] on the joint continuity of separately continuous bilinear operators are p
3 KB (545 words) - 12:08, 3 August 2014
...e of this generalized limit by using the [[Hahn–Banach
theorem|Hahn–Banach
theorem]]. Today (1996), Banach limits are studied via the notion of amenability.
...real numbers are amenable (left and right). M.M. Day has proved that every
Abelian semi-group is left and right amenable. On the other hand, <img align="absmi
10 KB (1,395 words) - 06:44, 9 October 2016
...[[Topological field|Topological field]]) satisfying the implicit function theorem, see [[#References|[a3]]].
...a1]</TD> <TD valign="top"> F.-V. Kuhlmann, "Valuation theory of fields, abelian groups and modules" , ''Algebra, Logic and Applications'' , Gordon&Brea
3 KB (405 words) - 19:33, 28 April 2014
...C ^ { * } ( G )$ and called the full $C ^ { * }$-algebra of $G$. If $G$ is Abelian and $\hat { C }$ its dual group, then $C ^ { * } ( G )$ is isometrically is
This Banach algebra is called the Fourier–Stieltjes algebra of $G$. If $G$ is Abelian, then $B ( G )$ is isometrically isomorphic to the Banach algebra of all bo
7 KB (1,059 words) - 15:30, 1 July 2020
...} )$, where $\hat { C }$ is the dual group of $G$. For $G$ not necessarily Abelian, $A _ { 2 } ( G )$ is precisely the [[Fourier-algebra(2)|Fourier algebra]]
...1]]]) a kind of "non-commutative harmonic analysis on G" , where (for $G$ Abelian) $A _ { p } ( G )$ replaces $L _ { \text{C} } ^ { 1 } ( \hat { G } )$ and $
11 KB (1,698 words) - 07:42, 27 January 2024
...nces|[1]]] and, in one special case, by F. Châtelet, that for an arbitrary Abelian variety $ A $
...trary orders [[#References|[4]]], [[#References|[5]]]. According to Lang's theorem, $ { \mathop{\rm WC} } ( A, k) = 0 $
7 KB (1,109 words) - 16:59, 1 July 2020
is only a [[Complex manifold|complex manifold]], but the Baily–Borel theorem [[#References|[a2]]] endows it with a canonical structure of a quasi-projec
...type), or Abelian motives with additional structure (Shimura varieties of Abelian type) [[#References|[a4]]], [[#References|[a6]]] (cf. also [[Moduli theory|
6 KB (890 words) - 19:42, 20 February 2021
...s , s ) ) )$. The main result of the theory is the arithmetic Riemann–Roch theorem, which computes the behaviour of the Chern character under direct image [[#
...nal points of $X$ is contained in the union of finitely many translates of Abelian proper subvarieties of $A$.
8 KB (1,219 words) - 21:00, 13 July 2020
...d relative homotopy groups as crossed modules, thus giving non-trivial non-Abelian information and often determining the $2$-type of a space. Some of the expl
...]]] for the presentation. This module should be thought of as giving a non-Abelian form of syzygies (cf. also [[Syzygy|Syzygy]]), and as the start of a free c
9 KB (1,326 words) - 16:58, 1 July 2020
the Hodge conjecture is equivalent to the [[Lefschetz theorem]] on cohomology of type $ ( 1, 1) $.
is a simple five-dimensional Abelian variety (see [[#References|[6]]]).
6 KB (935 words) - 09:01, 21 January 2024
...Maxwell equations|Maxwell equations]] (in vacuum). The quantization of non-Abelian gauge theories is still in its infancy.
...ohomology class. An analogous formula in dimension two is Gauss' classical theorem expressing the [[Euler characteristic|Euler characteristic]] as the integra
7 KB (1,013 words) - 16:45, 1 July 2020
theorem).
Other examples of group schemes are Abelian (group) varieties {{Cite|Mu}}.
5 KB (831 words) - 21:59, 5 March 2012
L. Stickelberger proved the following theorem: For $ r \geq 1 $,
Stickelberger's theorem implies that $ S $
7 KB (1,035 words) - 05:58, 19 March 2022
...m_{i+j=n} \binom{m_1}{i} \binom{m_2}{j}$ follows by the binomial expansion theorem from $(X+Y)^{m_1+m_2} = (X+Y)^{m_1} (X+Y)^{m_2}$.
.... A pre-$\lambda$-ring structure on $\Lambda(R)$ defines a homomorphism of Abelian groups $\lambda_t : R \rightarrow \Lambda(R)$, $\lambda_t(x) = \lambda^0(x)
10 KB (1,721 words) - 07:44, 23 March 2016
...operator whose norm does not exceed one) on a Hilbert space (von Neumann's theorem). This result is closely connected with the existence of a unitary power di
.... In the case when $\mathfrak A$ is the group algebra of a locally compact Abelian group, spectral sets are also called sets of harmonic synthesis.
2 KB (295 words) - 15:46, 29 December 2018
===Siegel's theorem on Dirichlet L-functions===
...lass number of a quadratic field of discriminant $-D$, it follows from the theorem that
5 KB (784 words) - 20:40, 18 October 2014
and an Abelian group $ G $.
By adding $ r $-dimensional chains as linear forms one obtains the Abelian group $ C _ {r} ( K, G) $
6 KB (897 words) - 08:54, 25 April 2022
but this is inessential.) Sheaves of Abelian groups, rings and other structures can be defined similarly.
Giraud's little theorem). Categories equivalent to one of the form $ Sh ( C, \tau ) $
8 KB (1,216 words) - 18:08, 14 November 2023
...cohomology theory" (but not the designation) while studying "generalized Abelian integrals" (now called "Eichler integrals" ; see below).
...idered here, a suitable version of the [[Riemann–Roch theorem|Riemann–Roch theorem]] shows that $C ^ { + } ( \Gamma , k , \mathbf{v} )$ has finite dimension o
13 KB (1,993 words) - 07:12, 15 February 2024
In formulation 1), the Lévy–Cramér theorem admits a generalization to the convolution of two signed measures with rest
...mér theorem to random variables in Euclidean spaces and in locally compact Abelian groups.
4 KB (647 words) - 19:21, 24 March 2023
...er of a group|Character of a group]]). Indeed, if $G$ is a locally compact Abelian group, the Fourier–Stieltjes transform of a finite measure $\mu$ on $\hat
...-definite functions on $G$. This definition is still valid when $G$ is not Abelian.
14 KB (2,163 words) - 19:56, 8 February 2024
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.
4 KB (596 words) - 15:30, 1 July 2020
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 $
6 KB (891 words) - 19:08, 26 March 2023
...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