...\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