significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. A result of Emil...

In mathematics, a Galois module is a G-module, with G being the Galois group of some extension of fields. The term Galois representation is frequently...

In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection...

group the Galois group of a Galois extension of the rational numbers? (more unsolved problems in mathematics) In Galois theory, the inverse Galois problem...

Galois theory is parallel to algebraic Galois theory. One difference between the two constructions is that the Galois groups in differential Galois theory...

In mathematics, Grothendieck's Galois theory is an abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the...

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any...

In mathematics, the absolute Galois group GK of a field K is the Galois group of Ksep over K, where Ksep is a separable closure of K. Alternatively it...

differential field extension generated by the solutions of a linear differential equation, using the differential Galois group of the field extension. A major goal...

Artin–Schreier theory is a branch of Galois theory, specifically a positive characteristic analogue of Kummer theory, for Galois extensions of degree equal to the characteristic...

(1984) and named by Matzat (1987). It appears in Galois theory, in the study of the inverse Galois problem or the embedding problem which is a generalization...

indeed Galois theory shows that this analogy is more than just a coincidence. The formula holds for both finite and infinite degree extensions. In the...

{Profinite groups}. The inverse Galois problem asks what groups can arise as fundamental groups (or Galois groups of field extensions). Anabelian geometry, for...

differential Galois theory, but this is not strictly true. The theorem can be proved without any use of Galois theory. Furthermore, the Galois group of a...

This improved statement follows directly from Galois theory § A non-solvable quintic example. Galois theory implies also that x 5 − x − 1 = 0 {\displaystyle...

differential Galois theory, a variant of Galois theory dealing with linear differential equations. Galois theory studies algebraic extensions of a field...

arithmetic dynamics, an arboreal Galois representation is a continuous group homomorphism between the absolute Galois group of a field and the automorphism...

vanishes. This is true for all finite Galois extensions of number fields, not just cyclic ones. For cyclic extensions the group H2(L/K) is isomorphic to...

inverse Galois problem for G {\displaystyle G} asks whether there exists a finite Galois extension F ∣ Q {\displaystyle F\mid \mathbb {Q} } whose Galois group...

"Teichmüller's Lego-game and the Galois group of Q over Q" ("Un jeu de “Lego-Teichmüller” et le groupe de Galois de Q sur Q"). 3. Number fields associated...

reasoning. Given a separable extension K′ of K, a Galois closure L of K′ is a type of splitting field, and also a Galois extension of K containing K′ that...

development of the Galois group. Since then it has been used in the development of Galois theory and the fundamental theorem of Galois theory. The primitive...

There also exists "finite level" modifications of the Galois groups: if E/F is a finite extension, then the relative Weil group of E/F is WE/F = WF/W c...

concerning jumps of the upper numbering filtration of the Galois group of a finite Galois extension. A special case of it when the residue fields are finite...

field extension [F(x) : F(y)]. This extension is generally not Galois but has Galois closure L(f). The associated Galois group of the extension [L(f) :...

characteristic p. The question addresses the existence of a Galois extension L of K(C), with G as Galois group, and with specified ramification. From a geometric...

shown that every cyclotomic field is an abelian extension of the rational number field Q, having Galois group of the form ( Z / n Z ) × {\displaystyle...

reciprocity. The Artin reciprocity law applies to a Galois extension of an algebraic number field whose Galois group is abelian; it assigns L-functions to the...

of integers modulo n and the Galois group of Q ( ω ) . {\displaystyle \mathbb {Q} (\omega ).} This shows that this Galois group is abelian, and implies...

is given the name, stating that if L/K is a finite Galois extension of fields with arbitrary Galois group G = Gal(L/K), then the first cohomology group...