In mathematics, in the 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. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them.
For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory.
area of abstract algebra known as Galois theory, the Galoisgroup 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...
absolute Galoisgroup GK of a field K is the Galoisgroup of Ksep over K, where Ksep is a separable closure of K. Alternatively it is the group of all automorphisms...
automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galoisgroup and obeys...
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...
In mathematics, a Galois module is a G-module, with G being the Galoisgroup of some extension of fields. The term Galois representation is frequently...
In mathematics, differential Galois theory studies the Galoisgroups of differential equations. Whereas algebraic Galois theory studies extensions of...
uses the fundamental theorem of Galois theory in specifying four intermediate fields between Q and T and their Galoisgroups, as well as two theorems on cyclic...
finite group the Galoisgroup of a Galois extension of the rational numbers? (more unsolved problems in mathematics) In Galois theory, the inverse Galois problem...
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...
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...
\omega ^{k}\right)} defines a group isomorphism between the units of the ring of integers modulo n and the Galoisgroup of Q ( ω ) . {\displaystyle \mathbb...
abstract algebra, an abelian extension is a Galois extension whose Galoisgroup is abelian. When the Galoisgroup is also cyclic, the extension is also called...
rise naturally to Galoisgroups that are profinite. Specifically, if L / K {\displaystyle L/K} is a Galois extension, consider the group G = Gal ( L /...
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...
In mathematics, the interplay between the Galoisgroup G of a Galois extension L of a number field K, and the way the prime ideals P of the ring of integers...
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...
In mathematics, a Weil group, introduced by Weil (1951), is a modification of the absolute Galoisgroup 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...
"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...
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