Finite extensions of local fields information

separable extensions of the residue field of K. Again, let L / K {\displaystyle L/K} be a finite Galois extension of nonarchimedean local fields with finite residue...

extension is a trivial extension. Extensions of degree 2 and 3 are called quadratic extensions and cubic extensions, respectively. A finite extension...

Non-Archimedean local fields of characteristic zero: finite extensions of the p-adic numbers Qp (where p is any prime number). Non-Archimedean local fields of characteristic...

mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which...

in the case of local fields with finite residue field and the idele class group in the case of global fields. The finite abelian extension corresponding...

The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic...

of number fields, function fields of algebraic curves over finite fields, and local fields. There are two slightly different definitions of the term cyclotomic...

generally involve different finite fields (for example the whole family of fields Z/pZ as p runs over all prime numbers). In these fields, the variable t is substituted...

field and let L be a finite Galois extension of K. Let Sv be the set of equivalence classes of extensions of v to L and let G be the Galois group of L...

cotangent sheaf Ω X / Y {\displaystyle \Omega _{X/Y}} is zero. Finite extensions of local fields Ramification (mathematics) Hartshorne 1977, Ch. IV, § 2. Grothendieck...

extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise...

such family of groups is the family of general linear groups over finite fields. Finite groups often occur when considering symmetry of mathematical...

every finite extension is separable. All fields of characteristic zero, and all finite fields, are perfect. Imperfect degree Let F be a field of characteristic...

every field extension F/k. (see below) Otherwise, k is called imperfect. In particular, all fields of characteristic zero and all finite fields are perfect...

global field is one of two types of fields (the other one is local fields) that are characterized using valuations. There are two kinds of global fields: Algebraic...

In mathematics, the classification of finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating...

concerns extensions of "local" (i.e., complete for a discrete valuation) fields with finite residue field.[dubious – discuss] Part I, Local Fields (Basic...

study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory. Given a field K, the...

cohomology of the idele class group vanishes. This is true for all finite Galois extensions of number fields, not just cyclic ones. For cyclic extensions the...

confined to an area of the body Local class field theory, the study of abelian extensions of local fields Local field, a special type of field that is a locally...

function assembling the information of the number of points of E with values in the finite field extensions Fpn of Fp. It is given by Z ( E ( F p ) , T...

cyclic cyclotomic extensions of fields; for finite fields they are given by the algebraic closure, for non-archimedean local fields they are given by...

\mathbb {Q} _{p}} and finite extensions K / Q p {\displaystyle K/\mathbb {Q} _{p}} . Each of these are examples of local fields. Note the algebraic closure...

notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory (since the root of any...