Galois extension information

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...

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...

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

all x in K. When the extension is Galois this automorphism group is called the Galois group of the extension. Extensions whose Galois group is abelian are...

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...

notation Gal(E/F) is only used for Galois extensions. If E/F is Galois, then Gal(E/F) = Aut(E/F). If E/F is not Galois, then the "correspondence" gives...

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...

algebraic extension to be a Galois extension. Bourbaki calls such an extension a quasi-Galois extension. For finite extensions, a normal extension is identical...

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...

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

C-extension of K. If E is a C-extension of F and E > K > F then E is a C-extension of K. Galois extension A normal, separable field extension. Galois group...

structure of the set 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...

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...

normal integral basis is part of Galois module theory. Let F ⊂ K {\displaystyle F\subset K} be a Galois extension with Galois group G {\displaystyle G} . The...

algebraic extension. The Galois group of N over M can again be defined as the group of automorphisms, and it turns out that most of the theory of Galois groups...

radical extension if and only if its Galois group is solvable (but there are non-radical Galois extensions whose Galois group is solvable, for example Q (...

of Galois theory, and, in particular, of the fundamental theorem of Galois theory. Let E ⊇ F {\displaystyle E\supseteq F} be an algebraic extension of...

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...

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

L/K} be a finite Galois extension of nonarchimedean local fields with finite residue fields ℓ / k {\displaystyle \ell /k} and Galois group G {\displaystyle...

{\dfrac {m\pi }{q}}\right),&q\equiv 1\mod 4.\end{cases}}} If K is a Galois extension of Q, the theory of Artin L-functions applies to ζ K ( s ) {\displaystyle...

mathematics, differential Galois theory studies the Galois groups of differential equations. Whereas algebraic Galois theory studies extensions of algebraic fields...

linear transformation. If L/K is a Galois extension, one may compute the norm of α ∈ L as the product of all the Galois conjugates of α: N L / K ⁡ ( α )...

algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground...