area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...

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

release from prison, Galois fought in a duel and died of the wounds he suffered. Galois was born on 25 October 1811 to Nicolas-Gabriel Galois and Adélaïde-Marie...

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 is the group of all automorphisms...

and roots similar to the formula above. Modern Galois theory generalizes the above type of Galois groups by shifting to field theory and considering field...

automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys...

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...

is a finite field extension of F whose Galois group is G. All subgroups and quotient groups of cyclic groups are cyclic. Specifically, all subgroups...

mathematics, differential Galois theory is the field that studies extensions of differential fields. Whereas algebraic Galois theory studies extensions...

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

development of Galois theory. In its most basic form, the theorem asserts that given a field extension E/F that is finite and Galois, there is a one-to-one...

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...

degree. Évariste Galois coined the term "group" and established a connection, now known as Galois theory, between the nascent theory of groups and field theory...

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

\omega ^{k}\right)} defines a group isomorphism between the units of the ring of integers modulo n and the Galois group of Q ( ω ) . {\displaystyle \mathbb...

mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups...

uses the fundamental theorem of Galois theory in specifying four intermediate fields between Q and T and their Galois groups, as well as two theorems on cyclic...

rise naturally to Galois groups that are profinite. Specifically, if L / K {\displaystyle L/K} is a Galois extension, consider the group G = Gal ⁡ ( L /...

abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called...

in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications...

satisfied if E has characteristic 0. For a finite Galois extension, the Galois group Gal(F/E) is the group of field automorphisms of F that are trivial on...

Évariste Galois in 1832, in his last letter (to Chevalier) and second (of three) attached manuscripts, which he used in the context of studying the Galois group...

"symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant...

Galois group; however in terms of the Tate conjecture and Galois representations on étale cohomology, it predicts the image of the Galois group, or, more...

In mathematics, the interplay between the Galois group G of a Galois extension L of a number field K, and the way the prime ideals P of the ring of integers...

In mathematics, a Weil group, introduced by Weil (1951), is a modification of the absolute Galois group of a local or global field, used in class field...

2007 preprint]{{citation}}: CS1 maint: postscript (link) Galois, Évariste (1846), "Lettre de Galois à M. Auguste Chevalier", Journal de Mathématiques Pures...

and to provide combinatorial invariants for the action of the absolute Galois group of the rational numbers. The name of these embeddings is French for a...

of extensions is known better when L/K is Galois. Let (K, v) be a valued field and let L be a finite Galois extension of K. Let Sv be the set of equivalence...

primes p and qi. The inertia group measures the difference between the local Galois groups at some place and the Galois groups of the involved finite residue...