...\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
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
3) In the category of Abelian groups, the functor $ \mathop{\rm Hom} ( A , Y ) $
...roups is the left adjoint of the functor of taking the torsion part of any Abelian group.
8 KB (1,301 words) - 20:14, 4 April 2020
...assigns the sum in the usual sense to any convergent series (an [[Abelian theorem]]). The series
2 KB (345 words) - 21:12, 23 November 2023
there is associated a sequence of Abelian groups $ H ^ { n } ( G, A) $,
is an Abelian group and $ G $
16 KB (2,427 words) - 09:48, 26 March 2023
are Abelian for all $ 0 \leq i < n $ );
is a one-dimensional (Abelian) Lie algebra for $ 0 \leq i < m $ .
9 KB (1,348 words) - 08:49, 8 April 2023
Given an arbitrary Abelian group $ \pi $
one can define a simplicial set (in fact, a simplicial Abelian group) $ E ( \pi , n) $.
32 KB (4,905 words) - 09:31, 13 February 2024
...his 1963 notes [[#References|[a7]]]. This facilitated a proof of a duality theorem of A. Grothendieck (cf. [[#References|[a5]]]). Let $ {\mathcal C} $
...seful to keep the example below in mind: the category of complexes over an Abelian category (and algebraic mapping cones, the corresponding long exact sequenc
16 KB (2,338 words) - 06:56, 10 May 2022
...orem of local [[Class field theory|class field theory]] there is for every Abelian extension of local fields $ K / k $
3 KB (398 words) - 08:27, 6 June 2020
...among all algebraic number fields is illustrated by the [[Kronecker–Weber theorem]], which states that a finite extension $ K/ \mathbf Q $
is Abelian if and only if $ K \subset K _ {n} $
12 KB (1,769 words) - 11:19, 26 March 2023
In the special case of the group algebra of a locally compact Abelian group (with convolution taken as multiplication in the algebra, cf. also [[
...[[#References|[a2]]], Sect. 11.13). A well-known such theorem is Wiener's theorem (cf. also [[#References|[a1]]], Chapt. XI, Sect. 2): If $ f ( t) = \sum _
3 KB (474 words) - 19:41, 5 June 2020
...ty|Rational variety]], [[Unirational variety|Unirational variety]]). Since Abelian varieties can never be rational, the main interest is in rationality theore
...ective). Fairly complete results are obtained for tori which split over an Abelian extension of the ground field (see [[#References|[5]]]). The first example
8 KB (1,072 words) - 20:22, 21 December 2019
...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
...are the three more extensive problems. The unidirectional word problem for Abelian semi-groups is decidable and, hence, so are the three subproblems described
...ility|Unsolvability]]; [[Gödel incompleteness theorem|Gödel incompleteness theorem]]. It was shown already by K. Gödel that the existence of undecidable prop
7 KB (1,138 words) - 13:14, 10 April 2014
...ative multiplication and addition exist; a near-ring is a (not necessarily Abelian) group with respect to addition, and the right distributive property
...$, the equality $A=M_S(\Gamma)$ holds (an analogue of the Wedderburn–Artin theorem). For every $\nu=0,1,2$, the Jacobson radical $J_\nu(A)$ of type $\nu$ can
5 KB (767 words) - 11:23, 27 October 2014
...s were introduced by J. Tate in order to study degenerations of curves and Abelian varieties over $ K $.
cf. also [[Weierstrass theorem|Weierstrass theorem]]); affinoid algebras are Noetherian rings, and even excellent rings if the
8 KB (1,161 words) - 08:25, 6 June 2020
...inable in an $o$-minimal expansion of $\mathbf{R}$" . This is a finiteness theorem, and van den Dries aims to explain the other finiteness phenomena in real a
2) An [[ordered group]] is $o$-minimal if and only if it is divisible Abelian.
5 KB (799 words) - 09:27, 26 November 2016
$#C+1 = 102 : ~/encyclopedia/old_files/data/D110/D.1100020 De Finetti theorem
...ent is De Finetti's theorem. Thus, an equivalent statement of De Finetti's theorem is that the extremal points of the convex set of exchangeable probability m
13 KB (1,888 words) - 11:23, 26 March 2023
...t})$. A theorem of Morita says that if $\mathcal{C}$ and $\mathcal{D}$ are Abelian full subcategories with $A \in \mathcal{C}$ and $B \in \mathcal{D}$, then a
3 KB (499 words) - 20:17, 26 November 2016
is a finite extension, then, according to the Chebotarev density theorem, for any automorphism $ \sigma \in \mathop{\rm Gal} ( K/k) $
For an Abelian extension $ K/k $,
4 KB (569 words) - 05:59, 11 October 2023
==Riesz decomposition theorem for super- or subharmonic functions.==
...nd [[Riesz theorem(2)|Riesz theorem]] (where it is simply called the Riesz theorem), [[#References|[a12]]], [[#References|[a20]]]. See also [[#References|[a8]
8 KB (1,180 words) - 17:00, 1 July 2020
$#C+1 = 96 : ~/encyclopedia/old_files/data/R081/R.0801980 Riemann\ANDRoch theorem
A theorem expressing the [[Euler characteristic|Euler characteristic]] $ \chi ( {\m
10 KB (1,385 words) - 03:10, 2 March 2022
...eory of compact groups, and all the results of that theory (the Peter–Weyl theorem, the theory of characters, orthogonality relations, etc.) are valid (and si
...ion of every irreducible representation is a divisor of the index of every Abelian normal subgroup of $ G $(
10 KB (1,488 words) - 19:39, 5 June 2020
Although the above theorem divides the Baumslag–Solitar groups into three classes: those that are re
...numbers. Thus, such groups are meta-Abelian (cf. [[Meta-Abelian group|Meta-Abelian group]]) and have strong structural properties; in particular, they do not
18 KB (2,803 words) - 16:46, 1 July 2020
...almost-periodic functions on a group depends essentially on the mean-value theorem (cf. [[#References|[5]]], [[#References|[8]]]). A linear functional $ M _
Theorem 1 (the Parseval equality). For an almost-periodic function $ f (x) $
12 KB (1,716 words) - 11:05, 10 May 2020
...from the category of pointed topological spaces into the category of (non-Abelian) groups. For any path $ \phi $
Poincaré's theorem).
6 KB (829 words) - 05:44, 13 April 2023
are coherent (the finiteness theorem). A similar fact holds for étale cohomology. In particular, if $ X $
the comparison theorem). 3) If $ X $
9 KB (1,267 words) - 08:08, 6 June 2020
...h unique root extraction. The following groups are orderable: torsion-free Abelian groups, torsion-free nilpotent groups, free groups, and free solvable group
...rderable groups is itself orderable. For orderable groups there is a local theorem (see [[Mal'tsev local theorems|Mal'tsev local theorems]]). A totally ordere
3 KB (495 words) - 19:18, 17 August 2014
...divisors corresponding to $ D $ (see [[Riemann–Roch theorem|Riemann–Roch theorem]]). If $ D $
is a free Abelian group of rank $ \rho $,
10 KB (1,443 words) - 15:43, 1 March 2022
is an Abelian group, then a transgression in $ E $
...fibre bundles. An important role is played here by the Borel transgression theorem: If $ A $
3 KB (536 words) - 09:05, 8 April 2023
...lian extension of $\textbf{Q}$, or, equivalently (by the [[Kronecker–Weber theorem]]), the maximal cyclotomic extension of $\textbf{Q}$.
...y, inverse problem of|Galois theory, inverse problem of]]). By the Iwasawa theorem [[#References|[a7]]], p. 567, (see also [[#References|[a1]]], Cor. 24.2), a
11 KB (1,686 words) - 18:54, 6 February 2021
even an Abelian group). By definition, if $ x = [ u ] $
even in the category of Abelian groups).
33 KB (4,910 words) - 10:04, 15 December 2019
...umbers without an obvious analogue for algebraic numbers is related to the theorem on unique factorization of rational integers $ n $ in prime factors: $$
...complicated. The question arises: What becomes of the unique factorization theorem, does it have a meaning at all in algebraic number fields?
28 KB (4,440 words) - 22:00, 11 December 2019
...presented to cover cases when an arbitrary locally compact [[Abelian group|Abelian group]] $G$ is represented by invertible operators $\{ U _ { t } \} _ { t \
...$\mathcal{I} \neq L ^ { 1 } ( G )$, i.e., if $x \neq 0$, by the Tauberian theorem (cf. also [[Tauberian theorems|Tauberian theorems]]). This hull is called t
14 KB (2,151 words) - 17:43, 1 July 2020
...nk, $ t _ \lambda = t ( H _ \lambda ( M) ) $ (the torsion rank of an Abelian group $ A $
...a Morse function for which the Morse inequalities are equalities (Smale's theorem, see [[#References|[2]]]). In particular, on any closed manifold that is ho
6 KB (845 words) - 01:33, 5 March 2022
be a free Abelian group with basis $ \{ e _ \omega \} = \{ e _ {i _ {1} \dots i _ {k} }
The cited theorem asserts that the homomorphism
8 KB (1,150 words) - 18:42, 13 January 2024
The theorem is also associated with the names of E.T. Whittaker, K. Ogura, V.A. Kotel'n
...ver, both situations are covered by the so-called "approximate sampling" theorem, which is valid for not necessarily band-limited signals. It is due to J.R.
9 KB (1,428 words) - 07:01, 19 March 2024
...ed on the [[Cauchy integral theorem|Cauchy integral theorem]]. The residue theorem is fundamental in this theory. Let $ f( z) $
in a neighbourhood of the point at infinity. The residue theorem implies the theorem on the total sum of residues: If $ f( z) $
16 KB (2,407 words) - 07:56, 11 January 2022
...s with $ \mathop{\rm End}\nolimits (V) $ ( "Burnside theoremBurnside's theorem" ). Every normal subgroup of a completely reducible linear group is complet
...explicit function of $ n $ (see also [[Lie–Kolchin theorem|Lie–Kolchin theorem]]); in particular, the commutator subgroup of $ H $ is a unipotent group
16 KB (2,362 words) - 18:01, 12 December 2019
...inear forms it is possible to introduce an addition, converting it into an Abelian group — the Witt group of $ k $(
cf. [[Witt theorem|Witt theorem]]), and also to the case of symmetric bilinear forms associated with quadra
8 KB (1,152 words) - 08:29, 6 June 2020
Then there is the following characterization theorem for the Witt vectors. There is a unique functor $ W : \ \mathbf{Ring} \r
be the Abelian group $ 1 + t A [[t]] $
17 KB (2,502 words) - 17:25, 22 December 2019
...theorem]], which was first obtained for Lie groups. A consequence of this theorem is the fact that every compact group admits a complete system of finite-dim
...here is the following fundamental structure theorem: Every locally compact Abelian group $ G $
14 KB (2,197 words) - 16:40, 31 March 2020
ramified at exactly four points; and also a one-dimensional Abelian
$X$ with the structure of an Abelian group that is compatible with the
19 KB (3,251 words) - 20:37, 19 September 2017
an Abelian group (or module over some ring) $ H _ {q} ( X, A) $
...(the uniqueness theorem). In the category of all polyhedra, the uniqueness theorem holds when the requirement is added that the homology (cohomology) of a uni
5 KB (779 words) - 19:51, 16 January 2024
''Hurwitz formula, Hurwitz theorem''
...n="top">[5]</TD> <TD valign="top"> S. Lang, "Introduction to algebraic and Abelian functions" , Addison-Wesley (1972) {{MR|0327780}} {{ZBL|0255.14001}} </TD><
4 KB (523 words) - 08:11, 6 June 2020
...quivalent to studying modules over the group algebra $KG$. Thus, Maschke's theorem is formulated in the language of group algebras as follows: If $G$ is a fin
...f an ordered group is imbeddable in a skew-field (the Mal'tsev–von Neumann theorem). It is believed that this is also true for any right-ordered group.
5 KB (872 words) - 22:00, 29 April 2012
...nected with the name of W. Burnside, who noted in 1897 that all simple non-Abelian groups which were known at that time were of even order [[#References|[1]]]
is an elementary Abelian group of order $ 2 ^ {d} $;
15 KB (2,211 words) - 08:50, 26 March 2023
form an Abelian group of order $ m $
form an Abelian group with respect to multiplication. The unit of this group is the class
20 KB (3,011 words) - 09:59, 26 March 2023
Cayley's theorem).
is a simple non-Abelian group. Hölder's theorem: For $ n \neq 2, 6 $,
7 KB (1,003 words) - 08:24, 6 June 2020
...ht modification of the same argument yields the so-called six exponentials theorem [[#References|[a2]]], [[#References|[a4]]]: If $x_1,x_2$ are two complex nu
...as results of algebraic independence on values of exponential, elliptic or Abelian functions; more generally it applies to the arithmetic study of commutative
5 KB (726 words) - 09:23, 20 December 2014
...cessarily free, J.R. Stallings [[#References|[a13]]] made use of Grushko's theorem [[#References|[a6]]], which asserts that if a [[Group|group]] $G$ is genera
In attempting to generalize the above Stallings' theorem to pairs of relative cohomological dimension one, C.T.C. Wall [[#References
9 KB (1,405 words) - 15:30, 1 July 2020
field), any finite extension is separable (the theorem on the
$\Gal(K/P)=H$. The main theorem in Galois theory states that these
11 KB (1,965 words) - 04:47, 16 January 2022
the Birkhoff–Frink theorem). Any lattice can be imbedded in the lattice $ \mathop{\rm Sub} A $
...c. The classic example of lattice definability is given by the first basic theorem of projective geometry (see [[#References|[1]]]), where vector spaces over
7 KB (1,037 words) - 08:24, 6 June 2020
constitutes a discrete Abelian group under addition, called the period group of $ f( z) $.
...group cannot consist of more than two basic independent periods (Jacobi's theorem).
5 KB (793 words) - 08:05, 6 June 2020
...logy spaces in dimensions $\geq1$. Also valid is the analogue of Grauert's theorem on the coherence of the image of a [[Coherent sheaf|coherent sheaf]] under
...D> <TD valign="top"> D. Mumford, "An analytic construction of degenerating abelian varieties over complete rings" ''Compos. Math.'' , '''24''' : 2 (1972) pp.
6 KB (892 words) - 20:07, 22 December 2018
Abelian) connected closed normal subgroups. The quotient group of a
The main classification theorem states that if $G'$ is another
4 KB (632 words) - 19:22, 12 December 2023
...cal stage of development of the theory of algebraic surfaces. Owing to the theorem on the resolution of singularities of algebraic surfaces, global methods of
...och theorem for algebraic surfaces. The generalization of the Riemann–Roch theorem for algebraic curves to algebraic surfaces is due to Castelnuovo (1897). Fo
26 KB (3,736 words) - 13:08, 8 February 2020
...origin of algebraic $K$-theory was an algebraic proof of the Riemann–Roch theorem
[[Bott periodicity theorem|Bott periodicity theorem]] — the theory of polynomial extensions — was obtained.
14 KB (2,405 words) - 22:14, 10 January 2015
of complex-valued regular Borel measures on a locally compact Abelian group $ G $
In the general case the theorem on idempotent measures can be naturally interpreted in terms of the cohomol
4 KB (655 words) - 13:07, 7 April 2023
...on" from Hasse's principle in the class of principal homogeneous spaces of Abelian varieties [[#References|[7]]], [[#References|[10]]]. The deviation is descr
which can be associated to each Abelian variety (the Tate–Shafarevich group). The principal difficulty of the the
24 KB (3,602 words) - 11:48, 26 March 2023
into an Abelian group with neutral element $ x _ {0} $.
A cubic curve endowed with this structure is a one-dimensional Abelian variety (an elliptic curve).
10 KB (1,376 words) - 11:12, 26 March 2023
....J. Jacobi on integrals of algebraic functions (see [[Abelian differential|Abelian differential]]). However, until recently, the only period mappings that hav
...ngularities of period mappings are described by the Schmid nilpotent orbit theorem, which, when $ S = \overline{S} \setminus \{ 0 \} $
12 KB (1,699 words) - 09:49, 20 December 2019
More exactly, let $D_0$ be a free Abelian semi-group with a unit element, the free generators of which are known as p
...t $K$ be the field of quotients of $A$, and let $D\supset D_0$ be the free Abelian group generated by the set of prime divisors. Then for any $c \in K^*$, $K^
16 KB (2,805 words) - 02:18, 6 January 2022
...with action of $G ( \overline { K } / K )$. For each $L$, let there be an Abelian extension $K ( L )$ of $K$ with the property that $K ( L ) \subset K ( L ^
...ed the main conjecture (using Kolyvagin's method) for all $F / \mathbf Q $ Abelian and all $p$, including $p = 2$.
19 KB (2,901 words) - 17:41, 25 November 2023
...erg–MacLane space|Eilenberg–MacLane space]]), where $\pi_n$ is some group (Abelian for $n > 1$). This system was introduced by M.M. Postnikov
The fundamental theorem in the theory of Postnikov systems states (see
11 KB (1,963 words) - 07:35, 8 August 2018
Abelian groups with respect to addition and semi-groups with respect
skew-field that is finite-dimensional over $F$ (Wedderburn's theorem);
11 KB (1,726 words) - 20:09, 15 December 2020
==Stability theorem.==
The stability theorem gives criteria for a valued function field to be a defectless field (cf. [[
13 KB (2,118 words) - 08:27, 6 June 2020
Abelian groups, one means a family of mappings (not necessarily homomorphisms) betw
forms an Abelian group with respect to the composition: $ ( \theta + \psi ) _ {X} = \theta
29 KB (4,197 words) - 09:49, 26 March 2023
...nce sets: For any multiplier of a $(v,k,\lambda)$-difference set $D$ in an Abelian group $G$ of order $v$ there exists a block in the block design determined
5 KB (832 words) - 10:37, 15 November 2014
is a free Abelian group $ J ^ \mu $
...tting the meridian equal to 1 (see below), property 2) follows from Hopf's theorem, according to which $ H ^ {2} ( G ; \mathbf Z ) $
12 KB (1,807 words) - 18:41, 13 January 2024
...ssence, that the classes of equivalent forms form a finite [[Abelian group|Abelian group]] with respect to composition.
...ed the [[Galois group|Galois group]] of the extension $K/k$. The principal theorem in [[Galois theory|Galois theory]] states that the mapping which associates
21 KB (3,246 words) - 17:24, 9 October 2016
...called the Steinberg symbol, as is also any symbol in any [[Abelian group|Abelian group]] $A$ for which c) and d) hold (which corresponds to a homomorphism o
8 KB (1,339 words) - 16:56, 1 July 2020
...ed most often in [[Universal algebra|universal algebra]], and in automatic theorem proving procedures.
Birkhoff's completeness theorem: If $S$ is an equational theory and $s = t$ is true in all the models of $S
9 KB (1,433 words) - 17:00, 1 July 2020
...two ways to reverse the bracketings as shown coincide. MacLane's coherence theorem says that then all other routes between two bracketed tensor products also
The coherence theorem for braided categories then asserts that different routes between tensor pr
17 KB (2,633 words) - 17:42, 1 July 2020
...m$-generator Burnside group $B ( m , 2 )$ of exponent $2$ is an elementary Abelian 2-group and the order $| B ( m , 2 ) |$ of $B ( m , 2 )$ is $2 ^ { m }$. Bu
...e presentations of $B ( m , n )$ constructed in [[#References|[a15]]], any Abelian or finite subgroup of $B ( m , n )$ was shown to be cyclic (for these and o
14 KB (2,317 words) - 06:45, 16 March 2023
...><td valign="top">[a6]</td> <td valign="top"> S. Shelah, "A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals"
5 KB (732 words) - 17:46, 1 July 2020
Every medial quasi-group is isotopic to an Abelian group $ Q ( \cdot ) $
Toyoda's theorem).
16 KB (2,475 words) - 19:16, 18 January 2024
...X _ { j }$ corresponding to arrows $\beta : i \rightarrow j$ of $Q$. By a theorem of L.A. Nazarova [[#References|[a12]]], given a connected quiver $Q$ the ca
...$\operatorname{rep}_K( Q )$. By the [[Jordan–Hölder theorem|Jordan–Hölder theorem]], the correspondence $\mathbf{X} \mapsto \underline{\operatorname { dim }}
18 KB (2,636 words) - 06:50, 15 February 2024
...lication in $L_1(G)$, then $L_1(G)$ becomes a Banach algebra; if $G$ is an Abelian locally compact group, then the Banach algebra $L_1(G)$ is commutative. The
...pointwise operation]]s), called the Fourier algebra of the locally compact Abelian group $\hat{G}$. In particular, if $G$ is the group of integers $\Z$, then
14 KB (2,346 words) - 22:48, 29 November 2014
These algebras were introduced by J. von Neumann . According to a theorem of von Neumann, a self-adjoint subalgebra $ A \subset {\mathcal B} ( H)
majorizes a non-zero Abelian projection. (An Abelian projection is a projection $ e $
17 KB (2,631 words) - 19:37, 19 January 2024
...lk-wise algebraic operations and the projection $p$ is called the sheaf of Abelian groups (rings, etc.) over $X$ associated with the pre-sheaf $F$.
...y of sheaves over $X$ has the same classical properties as the category of Abelian groups or the category of modules; in particular, one can define for sheave
26 KB (4,342 words) - 15:06, 15 July 2014
is a ring of polynomials in infinitely many indeterminates (Lazard's theorem).
...ism|cobordism]] is a universal one-dimensional formal group law (Quillen's theorem) and its logarithm is given by Mishchenko's formula$$
17 KB (2,537 words) - 22:38, 15 December 2019
[[Witt theorem|Witt theorem]]). Multiplying $f$ by a suitable scalar, one can, without changing the uni
...mples of unitary groups having an infinite series of normal subgroups with Abelian factors, examples of unitary groups for which $n=2$ and $\U_n^+(K,f)$ does
6 KB (1,075 words) - 16:59, 30 November 2014
equivalently, the category of (left) $ A $-modules, is an Abelian category. Note that if $ e $
...]] of all finite-length representations modulo exact sequences is the free Abelian group on the set of isomorphism classes of simple representations. A repres
22 KB (3,137 words) - 20:01, 15 March 2023
...functors from the category of pairs of spaces into the category of graded Abelian groups.
of graded Abelian groups (that is, to each pair of spaces $ ( X, A ) $
23 KB (3,297 words) - 19:41, 5 June 2020
an Abelian group $ H _ {r} ( X, A) $;
...in the theorems. On the other hand, instead of taking the category of all Abelian groups as the range of $ H _ {r} $,
23 KB (3,393 words) - 08:51, 25 April 2022
Thus there is a complete theory of finite Abelian groups (see [[Abelian group|Abelian group]]) and various classes of finite $ p $-
...) in non-solvable groups — an alternative formulation of the Feit–Thompson theorem — and the expression of properties of simple groups in terms of centraliz
12 KB (1,742 words) - 09:10, 4 July 2020
...meter group of unitary operators $\{U_t\}$ (here $U_t=e^{itA}$, by Stone's theorem).
|valign="top"|{{Ref|P}}|| W. Parry, "Compact abelian group extensions of discrete dynamical systems" ''Z. Wahrsch. verw. Geb.''
5 KB (710 words) - 12:58, 16 July 2014
...roblems related to the recursive presentability of models, Ershov's kernel theorem turns out to be useful. Its application to concrete algebraic systems allow
A large class of recursively presented models is given by the following theorem: Any countable model of an $ \aleph _ {1} $-
14 KB (1,981 words) - 08:10, 6 June 2020
...tail in the 19th century concerned Abelian functions, which are related to Abelian varieties in a way similar to the relationship between elliptic functions a
...hibiting domains $ D $ and groups $ \Gamma $ for which the following theorem on algebraic relations is true. If $ f _{1}, \dots, f _{n} $ are algebra
17 KB (2,502 words) - 06:16, 12 July 2022
==Hilbert's basis theorem==
...bert's theorem on invariants (see below, 8). Subsequently, Hilbert's basis theorem was extensively used in commutative algebra.
18 KB (2,720 words) - 19:17, 19 December 2019
...operatorname { Tor } _ { 1 } ^ { B } ( T , Y ) = 0 \}$. The Brenner–Butler theorem says that the functors $\operatorname{Hom}_H( T , - )$, respectively $T \ot
...ernshtein, I.M. Gel'fand and V.A. Ponomarev for their proof of the Gabriel theorem [[#References|[a4]]].
8 KB (1,215 words) - 19:50, 24 December 2023
...ors), in the theory of complex spaces, in the theory of categories (topoi, Abelian categories), and in functional analysis (representation theory). Conversely
...estigate the group of homomorphisms of a one-dimensional [[Abelian variety|Abelian variety]].
29 KB (4,414 words) - 17:20, 17 December 2019
...portant in the extension of the [[Kronecker theorem|Kronecker theorem]] on Abelian fields for imaginary quadratic ground fields and, more generally, in [[Clas
7 KB (1,010 words) - 19:41, 5 June 2020
...result on the possibility of such a simultaneous immersion is Matsusaka's theorem. Then the difficult problem remains of the existence of the quotient $ H /
...is route the existence of coarse moduli spaces has been proved for curves, Abelian varieties and $ K 3 $ -
16 KB (2,402 words) - 11:49, 16 December 2019
...}$. One assumes that this pairing is non-degenerate. The Stone–von Neumann theorem asserts that $H$ has a unique [[Irreducible representation|irreducible repr
...f. also [[Automorphism|Automorphism]]). If $g\in G$, the Stone–von Neumann theorem implies that $\Box\:\pi\cong\pi$. Let $\omega(g):\pi\to\Box$ be an intertwi
7 KB (1,158 words) - 20:32, 13 March 2024
...Extremally-disconnected space]]). It follows from the Stone representation theorem for Boolean algebras (cf. also [[Boolean algebra|Boolean algebra]]) that th
...2$, see [[#References|[a18]]]. As a corollary of the Sullivan–Weiss–Wright theorem [[#References|[a19]]], the Takenouchi and Dyer factors are isomorphic. Much
14 KB (2,118 words) - 16:46, 1 July 2020
...ups of this category, the homology and cohomology groups with values in an Abelian group, etc.
of complex numbers the following comparison theorem is valid: The groups $ \pi _{i} (X) $
6 KB (842 words) - 13:07, 15 December 2019
...enheim–Skolem theorem (cf. [[Gödel completeness theorem|Gödel completeness theorem]]) guarantees that each theory has a model in each infinite cardinality. On
M.D. Morley began this process with his 1963 generalization of a theorem of E. Steinitz by showing that for any first-order theory $ T $(
11 KB (1,671 words) - 11:38, 22 December 2019
of H. Cartan (cf. [[Cartan theorem|Cartan theorem]]) hold:
dimensional cell complex. On the other hand, for any countable Abelian group $ G $
10 KB (1,513 words) - 08:23, 6 June 2020
...eenrod–Sitnikov homology theory $H _ { * } ^ { S } (\cdot \ ; G )$ ($G$ an Abelian group) J. Milnor [[#References|[a3]]] established the following axiomatic c
Milnor's uniqueness theorem admits an extension for generalized homology theories [[#References|[a1]]]:
8 KB (1,134 words) - 17:46, 1 July 2020
This theorem easily implies that subalgebras and quotient algebras (with respect to clos
is finite. P. Civin and B. Yood had proved this for Abelian groups in [[#References|[a3]]]. In [[#References|[a12]]] it is shown that t
8 KB (1,127 words) - 07:28, 26 March 2023
for all Abelian sheaves $ {\mathcal F} $
By Serre's theorem, $ \mathop{\rm cohcd} ( X) = 0 $
9 KB (1,321 words) - 17:45, 4 June 2020
is a totally ordered Abelian group, the adjoined element $ \infty $
For each totally ordered Abelian group $ \Gamma $
14 KB (2,135 words) - 08:27, 6 June 2020
is a locally constant sheaf of finitely-generated Abelian groups on $ {\mathcal S} $,
is quasi-unipotent [[#References|[a3]]] (monodromy theorem). A polarized variation of Hodge structure over $ S $
6 KB (883 words) - 18:39, 2 January 2021
...ule]]) and any lattice of subobjects of an object in an [[Abelian category|Abelian category]].
...n algebras (cf. [[Boolean algebra|Boolean algebra]]). The coordinatization theorem of [[Projective geometry|projective geometry]] implies that any Arguesian r
29 KB (4,201 words) - 16:31, 9 December 2023
...by $\| \hat { f } \| = \| f \| _ { 1 }$; here, $G$ is any locally compact Abelian group, $\hat { C }$ is its dual group, and $\hat { f }$ is the Fourier tran
...ill holds for certain Beurling algebras. Secondly, the following injection theorem for Ditkin sets holds: If $\Gamma$ is a closed subgroup of $\hat { C }$, an
10 KB (1,614 words) - 16:56, 1 July 2020
...topology..." [[#References|[a1]]], "...a principle or yoga rather than a theorem" [[#References|[a5]]], and "...a commonplace of experience among topologi
Any notion (definition, theorem, etc.) in a [[Category|category]] $C$ which can be expressed purely categor
13 KB (1,969 words) - 08:10, 15 February 2024
.... In the non-compact case, Spector's proof involves a deep new fixed-point theorem for not necessarily affine mappings preserving a compact convex subset in a
...l–Nardzewski fixed-point theorem [[#References|[a7]]], embodying Spector's theorem.
9 KB (1,358 words) - 22:11, 5 June 2020
the kernels of which are the Abelian Lie algebra $ V $
...tsev decomposition|Levi–Mal'tsev decomposition]]) for Lie algebras with an Abelian radical [[#References|[1]]], [[#References|[5]]], [[#References|[14]]]. If
21 KB (3,027 words) - 17:45, 4 June 2020
...es of size $r$ over $k$). The tensor multiplication of algebras induces an Abelian group structure on the set of equivalence classes of finite-dimensional cen
...aically closed field of constants, then its Brauer group is zero ([[Tsen's theorem]]). The case of an arbitrary field of constants is treated in {{Cite|Fa}} a
7 KB (1,232 words) - 12:12, 30 December 2015
...tion and a multiplication is called a ring if: 1) it is an [[Abelian group|Abelian group]] with respect to addition (in particular, the ring has a zero elemen
...ing such algebras is the [[Wedderburn–Mal'tsev theorem|Wedderburn–Mal'tsev theorem]] "on the splitting of a radical" . It concerns the decomposition of a fini
21 KB (3,225 words) - 09:25, 13 July 2022
mappings, constitutes an Abelian group with respect to the so-called track addition [[#References|[1]]], [[#
...is theory include Hurewicz's isomorphism theorem and Hopf's classification theorem. $ D _ { n } $
11 KB (1,442 words) - 13:45, 8 June 2020
...iate section for this $\Spin_\C$ structure. Note that $A$ is just a $U(1)$ Abelian connection, and so $F = dA$, with $F^+$ being the self-dual part of $F$.
...\psi)$ is the trivial one. This means that one has a new kind of vanishing theorem in four dimensions ([[#References|[a1]]], 1994): No four-dimensional manifo
16 KB (2,663 words) - 10:57, 13 February 2024
with Abelian coefficient group $ G $,
...nder duality]]). A special case of these dualities is the Steenrod duality theorem (see [[Duality|Duality]] in algebraic topology), since the Kolmogorov duali
10 KB (1,511 words) - 16:36, 13 January 2024
the Brouwer–Hopf theorem); this isomorphism relates an element of the group $ \pi _ {n} ( S ^ {n}
there are no elements of odd Hopf invariant (as was known long before this theorem was proved, its assertion is equivalent to the following Frobenius conjectu
33 KB (4,248 words) - 08:17, 21 March 2022
...gories are additive (cf. [[Additive category|Additive category]]), but not Abelian. Isomorphisms in them are called topological isomorphisms; these are linear
...y if their matrices coincide up to a permutation of the Jordan blocks, the theorem means that Jordan matrices present, after suitable identification, a comple
67 KB (9,247 words) - 17:12, 29 October 2017
...thogonal projection given by the [[Cauchy integral theorem|Cauchy integral theorem]]. The [[C*-algebra|$C ^ { * }$-algebra]] ${\cal T} ({\bf T} ) : = C ^ { *
...pact operators; in fact, this "Toeplitz extension" is the generator of the Abelian group $\operatorname{Ext} ( {\cal C } ( {\bf T } ) ) \approx \bf Z$.
8 KB (1,186 words) - 16:46, 1 July 2020
...ery natural number is a sum of four squares of natural numbers (Lagrange's theorem), i.e. the sequence of squares is a basis of $ \mathbf Z _ {0} $
Hilbert's theorem), the order of which has been estimated by the [[Vinogradov method|Vinograd
28 KB (4,564 words) - 07:37, 26 March 2023
...ential equations includes the theory of algebraic functions, the theory of Abelian integrals, the theory of special functions, etc. Special functions — [[Be
Cauchy's theorem: Let the function $ f(t,\ x) $
16 KB (2,410 words) - 11:15, 28 January 2020
..., for example). According to the general Karhunen spectral representation theorem, a spectral representation (1) exists for a random function $ X ( t) $
...). The existence of such a spectral decomposition follows from the general theorem of Khinchin (or Wiener–Khinchin) on the integral representation of the co
17 KB (2,406 words) - 20:00, 12 January 2024
is an Abelian ideal in $ g $.
of compact type is solved by the following theorem.
9 KB (1,321 words) - 19:42, 5 June 2020
be an Abelian group, and let $ k \geq 1 $
...lved in a classical formulation in 1969 (A.T. Fomenko), when the following theorem was proved: If one is given a $ ( k- 1) $-dimensional submanifold $ \Ga
10 KB (1,540 words) - 11:46, 3 March 2022