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Longer titles found: Absolute Galois group (view), Shafarevich's theorem on solvable Galois groups (view)

searching for Galois group 62 found (279 total)

Class field theory (2,212 words) [view diff] exact match in snippet view article find links to article

and writing K for the maximal abelian unramified extension of F, the Galois group of K over F is canonically isomorphic to the ideal class group of F.

Picard–Vessiot theory (898 words) [view diff] exact match in snippet view article find links to article

solutions of a linear differential equation, using the differential Galois group of the field extension. A major goal is to describe when the differential

Grothendieck–Katz p-curvature conjecture (838 words) [view diff] exact match in snippet view article find links to article

this conjecture is essentially the same as saying that the differential Galois group G (or strictly speaking the Lie algebra g of the algebraic group G, which

Semilinear map (2,059 words) [view diff] exact match in snippet view article find links to article

transform; formally, the semidirect product of a linear group with the Galois group of field automorphism. For example, PΣU is used for the semilinear analogs

Grothendieck's Galois theory (569 words) [view diff] exact match in snippet view article find links to article

regarding Z ^ {\displaystyle {\hat {\mathbb {Z} }}} as the profinite Galois group Gal(F/F) of the algebraic closure F of any finite field F, over F. That

Cubic equation (10,290 words) [view diff] exact match in snippet view article find links to article

{\sqrt {\Delta }}} is fixed by the Galois group only if the Galois group is A3. In other words, the Galois group is A3 if and only if the discriminant

Equivariant L-function (159 words) [view diff] exact match in snippet view article find links to article

associated with it, corresponding to the characters of representations of the Galois group. By contrast, each extension has a unique corresponding equivariant L-function

Cyclic (mathematics) (259 words) [view diff] exact match in snippet view article

decomposition (group theory) Cyclic extension, a field extension with cyclic Galois group Graph theory: Cyclic function, a periodic function Cycle graph, a connected

Bauerian extension (283 words) [view diff] exact match in snippet view article find links to article

is the Galois extension of Q by the roots of 2x5 − 32x + 1, which has Galois group S5. Splitting of prime ideals in Galois extensions Narkiewicz (1990)

Valentiner group (523 words) [view diff] exact match in snippet view article find links to article

Valentiner group as a Galois group, and gave an order 3 differential equation with the Valentiner group as its differential Galois group. Coble, Arthur B.

Inner form (438 words) [view diff] exact match in snippet view article find links to article

whether a group is an inner or outer form one looks at the action of the Galois group G a l ( K ¯ / K ) {\displaystyle \mathrm {Gal} ({\overline {K}}/K)} on

Satake diagram (911 words) [view diff] exact match in snippet view article find links to article

the Galois group, with the simple roots vanishing on S colored black. In the case when k is the field of real numbers, the absolute Galois group has order

Brauer–Nesbitt theorem (439 words) [view diff] exact match in snippet view article find links to article

(continuous) l {\displaystyle l} -adic representations of the absolute Galois group of some field K {\displaystyle K} , unramified outside some finite set

Hilbert's irreducibility theorem (732 words) [view diff] exact match in snippet view article find links to article

immediately implies that if a finite group G can be realized as the Galois group of a Galois extension N of E = Q ( X 1 , … , X r ) , {\displaystyle E=\mathbb

Finite extensions of local fields (371 words) [view diff] exact match in snippet view article find links to article

fields with finite residue fields ℓ / k {\displaystyle \ell /k} and Galois group G {\displaystyle G} . Then the following are equivalent. (i) L / K {\displaystyle

Algebraic torus (3,965 words) [view diff] exact match in snippet view article find links to article

separable closure. This induces canonical continuous actions of the absolute Galois group of K on the lattices. The weights and coweights that are fixed by this

Main conjecture of Iwasawa theory (1,108 words) [view diff] exact match in snippet view article find links to article

Galois group of F∞ isomorphic to the p-adic integers. γ is a topological generator of Γ Ln is the p-Hilbert class field of Fn. Hn is the Galois group

Fiber functor (789 words) [view diff] exact match in snippet view article find links to article

IV Motivic Galois group - https://web.archive.org/web/20200408142431/https://www.him.uni-bonn.de/fileadmin/him/Lecture_Notes/motivic_Galois_group.pdf

List of permutation topics (280 words) [view diff] exact match in snippet view article find links to article

(permutation group theory) Cayley's theorem Cycle index Frobenius group Galois group of a polynomial Jucys–Murphy element Landau's function Oligomorphic group

Serre group (419 words) [view diff] exact match in snippet view article find links to article

module of characters, a finite free Z-module with an action of the finite Galois group Gal(L/Q). If L* is the algebraic group with L*(A) the units of A⊗L, then

Modular curve (2,016 words) [view diff] exact match in snippet view article find links to article

edges (black and white dots) are the points lying over 0 and 1. The Galois group of the covering X(7) → X(1) is a simple group of order 168 isomorphic

Schwarz's list (904 words) [view diff] exact match in snippet view article find links to article

differential Galois group of the hypergeometric equation is a solvable group. A general result connecting the differential Galois group G and the monodromy

Algebraic differential equation (429 words) [view diff] exact match in snippet view article find links to article

theory the case of algebraic solutions is that in which the differential Galois group G is finite (equivalently, of dimension 0, or of a finite monodromy group

Thaine's theorem (292 words) [view diff] exact match in snippet view article find links to article

dividing p − 1 {\displaystyle p-1} . Let G + {\displaystyle G^{+}} be the Galois group of F = Q ( ζ p + ) {\displaystyle F=\mathbb {Q} (\zeta _{p}^{+})} over

Generalized Riemann hypothesis (1,318 words) [view diff] exact match in snippet view article find links to article

Chebotarev density theorem: if L/K is a finite Galois extension with Galois group G, and C a union of conjugacy classes of G, the number of unramified

List of irreducible Tits indices (1,109 words) [view diff] exact match in snippet view article find links to article

close to each other (the orbit of the vertices under the *-action of the Galois group of k) and with certain sets of vertices circled (the orbits of the non-distinguished

Jean-Marc Fontaine (491 words) [view diff] exact match in snippet view article find links to article

He introduced the concept of geometric Galois representation of the Galois group of a number field. He also worked on Bloch-Kato conjectures. In 1984

Rational variety (1,547 words) [view diff] exact match in snippet view article find links to article

theory she studied the problem of parameterizing the equations with given Galois group, which she reduced to "Noether's problem". (She first mentioned this

Quadratic field (1,288 words) [view diff] exact match in snippet view article find links to article

theory, there being a unique subgroup of index 2 {\displaystyle 2} in the Galois group over Q {\displaystyle \mathbf {Q} } . As explained at Gaussian period

Mumford–Tate group (814 words) [view diff] exact match in snippet view article find links to article

conjecture, the Mumford–Tate group has been connected to the motivic Galois group, and, for example, the general issue of extending the Sato–Tate conjecture

History of group theory (3,565 words) [view diff] exact match in snippet view article find links to article

substitutions of the group. In modern terms, the solvability of the Galois group attached to the equation determines the solvability of the equation with

Ray class field (773 words) [view diff] exact match in snippet view article find links to article

of K associated to a ray class group by class field theory, and its Galois group is isomorphic to the corresponding ray class group. The proof of existence

Hecke character (1,738 words) [view diff] exact match in snippet view article find links to article

theory identifies the Hilbert characters with the characters of the Galois group of the Hilbert class field. For the field of rational numbers, the idele

Abstract algebra (4,185 words) [view diff] exact match in snippet view article find links to article

Lagrange's 1770 study of the solutions of the quintic equation led to the Galois group of a polynomial. Gauss's 1801 study of Fermat's little theorem led to

Tannakian formalism (834 words) [view diff] exact match in snippet view article find links to article

The closely-related algebraic groups Mumford–Tate group and motivic Galois group arise from categories of Hodge structures, category of Galois representations

Projective linear group (5,611 words) [view diff] exact match in snippet view article find links to article

the general linear group over a prime field, GL(ν, p), in studying the Galois group of the general equation of degree pν. The groups PSL(n, q) (general n

Conductor of an elliptic curve (1,006 words) [view diff] exact match in snippet view article find links to article

curve of order l for a prime l, P is the Swan representation, and G the Galois group of a finite extension of K such that the points of M are defined over

Quasi-algebraically closed field (1,067 words) [view diff] exact match in snippet view article find links to article

algebraically closed field is weakly C1. Any field with procyclic absolute Galois group is weakly C1. Any field of positive characteristic is weakly C2. If the

Moss Sweedler (737 words) [view diff] exact match in snippet view article find links to article

idempotent cohomology classes and algebras by partially ordered sets with a Galois group action". Amer. J. Math. 105 (3): 689–814. doi:10.2307/2374320. JSTOR 2374320

Lubin–Tate formal group law (1,101 words) [view diff] exact match in snippet view article find links to article

gfn-1(t)=g(f(f(⋯(f(t))⋯))). Then K(θn) is an abelian extension of K with Galois group isomorphic to U/1+pn where U is the unit group of the ring of integers

Newton's identities (7,642 words) [view diff] exact match in snippet view article find links to article

field whose Galois group permutes them according to the full symmetric group, and the field fixed under all elements of the Galois group is the base field)

Drinfeld module (1,623 words) [view diff] exact match in snippet view article find links to article

automorphic representations of GLn and certain representations of a Galois group. Drinfeld used Drinfeld modules to prove some special cases of the Langlands

Yves André (581 words) [view diff] exact match in snippet view article find links to article

overview". YouTube. 11 November 2016. "Yves André: What is... a motivic Galois group". YouTube. 18 January 2018. "Yves André: Periods of relative 1 motives"

List of number fields with class number one (1,708 words) [view diff] exact match in snippet view article find links to article

Murty showed that of all CM fields whose Galois closure has solvable Galois group, only finitely many have class number 1. A complete list of the 172 abelian

Uwe Jannsen (497 words) [view diff] exact match in snippet view article find links to article

In the 1980s with Kay Wingberg he completely described the absolute Galois group of p-adic number fields, i.e. in the local case. In 1994 he was an Invited

Artin–Schreier curve (1,078 words) [view diff] exact match in snippet view article find links to article

{\displaystyle p} . Such a cover is necessarily cyclic, that is, the Galois group of the corresponding algebraic function field extension is the cyclic

Big Finish Short Trips (754 words) [view diff] case mismatch in snippet view article find links to article

Jennah Dean Tenth Doctor (Ayesha Antoine) February 2023 (2023-02) 6 "The Galois Group" Scott Handcock Felicia Barker Eleventh Doctor, Valarie (Safiyya Ingar)

Alexander Grothendieck (8,253 words) [view diff] exact match in snippet view article find links to article

Topoi Étale cohomology and l-adic cohomology Motives and the motivic Galois group (Grothendieck ⊗-categories) Crystals and crystalline cohomology, yoga

Linear algebraic group (6,000 words) [view diff] exact match in snippet view article find links to article

finite type over k.) For example, the Mumford–Tate group and the motivic Galois group are constructed using this formalism. Certain properties of a (pro-)algebraic

Integral element (5,304 words) [view diff] exact match in snippet view article find links to article

{\displaystyle {\mathfrak {p}}_{1}\cap A\neq {\mathfrak {p}}_{2}\cap A} . The Galois group Gal ⁡ ( L / K ) {\displaystyle \operatorname {Gal} (L/K)} then acts on

Glossary of module theory (2,610 words) [view diff] exact match in snippet view article find links to article

reciprocity. Galois A Galois module is a module over the group ring of a Galois group. generating set A subset of a module is called a generating set of the

Torsor (algebraic geometry) (2,638 words) [view diff] exact match in snippet view article

{\displaystyle \operatorname {Gal} (L/K)} -torsor (roughly because the Galois group acts simply transitively on the roots.) By abuse of notation we have

Safiyya Ingar (329 words) [view diff] case mismatch in snippet view article find links to article

Chronicles (multiple boxed sets) Valarie Lockwood 2022 The War Master / Escape From Reality Hobgoblin 2023 Short Trips: The Galois Group Valarie Lockwood

Loewy decomposition (7,132 words) [view diff] exact match in snippet view article find links to article

irreducible it may occur that its Galois group is nontrivial, then algebraic solutions may exist. If the Galois group is trivial it may be possible to

List of Doctor Who audio plays by Big Finish (4,261 words) [view diff] case mismatch in snippet view article find links to article

Jennah Dean Tenth Doctor (Ayesha Antoine) February 2023 (2023-02) 6 "The Galois Group" Scott Handcock Felicia Barker Eleventh Doctor, Valarie (Safiyya Ingar)

List of examples of Stigler's law (4,945 words) [view diff] exact match in snippet view article find links to article

is often attributed to Georges-Louis Leclerc. Frobenius elements in a Galois group of global fields were first created by Dedekind. Fibonacci numbers. Fibonacci

Adele ring (18,442 words) [view diff] exact match in snippet view article find links to article

local class field theory gives a homomorphism of the idele group to the Galois group of the maximal abelian extension of the global field. The Artin reciprocity

Ludvig Sylow (2,698 words) [view diff] no match in snippet view article find links to article

Broch's absence abroad, during which he also began to treat and lecture Galois' group theory. But instead, his career simply stopped. When Broch again became

Witt vector (7,389 words) [view diff] exact match in snippet view article find links to article

which characterizes the real closed fields as those whose absolute Galois group has order two. This inspired them to ask what other fields had finite

Moy–Prasad filtration (4,086 words) [view diff] exact match in snippet view article find links to article

and then taking the fixed points of this Moy–Prasad group under the Galois group of k nr {\displaystyle k^{\text{nr}}} over k {\displaystyle k} . The

Faithfully flat descent (2,270 words) [view diff] exact match in snippet view article find links to article

a\mapsto \sigma (a)v} where the product runs over the elements in the Galois group of F / k {\displaystyle F/k} . An étale descent is a consequence of a

Artin transfer (group theory) (28,815 words) [view diff] exact match in snippet view article

goes back to 1934, when A. Scholz and O. Taussky tried to determine the Galois group G = G 3 ∞ ( K ) = G a l ( F 3 ∞ ( K ) | K ) {\displaystyle G=\mathrm