In mathematics, particularly in algebra, a field extension is a pair of fields K ⊆ L {\displaystyle K\subseteq L} , such that the operations of K are...

finite extension of a finite field is a cyclic extension. Class field theory provides detailed information about the abelian extensions of number fields, function...

mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important...

In field theory, a branch of algebra, an algebraic field extension E / F {\displaystyle E/F} is called a separable extension if for every α ∈ E {\displaystyle...

mathematics, an algebraic extension is a field extension L/K such that every element of the larger field L is algebraic over the smaller field K; that is, every...

In algebra, a purely inseparable extension of fields is an extension k ⊆ K of fields of characteristic p > 0 such that every element of K is a root of...

mathematics, a transcendental extension L / K {\displaystyle L/K} is a field extension such that there exists an element in the field L {\displaystyle L} that...

mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the...

In field theory, a simple extension is a field extension that is generated by the adjunction of a single element, called a primitive element. Simple extensions...

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...

In abstract algebra, a normal extension is an algebraic field extension L/K for which every irreducible polynomial over K that has a root in L splits...

symmetries of field extensions, provides an elegant proof of the Abel–Ruffini theorem that general quintic equations cannot be solved in radicals. Fields serve...

then E is an extension field of F. We then also say that E/F is a field extension. Degree of an extension Given an extension E/F, the field E can be considered...

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...

specifically in field theory, a radical extension of a field K {\displaystyle K} is a field extension obtained by a tower of field extensions, each generated...

In field theory, a branch of mathematics, the minimal polynomial of an element α of an extension field of a field is, roughly speaking, the polynomial...

theory of extension, in geometry Field extension, in Galois theory Group extension, in abstract algebra and homological algebra Homotopy extension property...

applied in the context of a finite field extension L/K, by using the field trace. This requires the property that the field trace TrL/K provides a non-degenerate...

mathematics, an algebraic number field (or simply number field) is an extension field K {\displaystyle K} of the field of rational numbers Q {\displaystyle...

xn − 1. A field extension that is contained in an extension generated by the roots of unity is a cyclotomic extension, and the extension of a field generated...

abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial splits...

be a field extension of K. An extension of v (to L) is a valuation w of L such that the restriction of w to K is v. The set of all such extensions is studied...

a subfield. Let K be a field and L a finite extension (and hence an algebraic extension) of K. The field L is then a finite-dimensional vector space over...

the field trace is a particular function defined with respect to a finite field extension L/K, which is a K-linear map from L onto K. Let K be a field and...

specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local field extension, which gives detailed information...

algebraic function field (often abbreviated as function field) of n variables over a field k is a finitely generated field extension K/k which has transcendence...

finite residue field. Let L / K {\displaystyle L/K} be a finite Galois extension of nonarchimedean local fields with finite residue fields ℓ / k {\displaystyle...

particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures...

mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local...

Agricultural extension is the application of scientific research and new knowledge to agricultural practices through farmer education. The field of 'extension' now...