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...

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, 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...

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, 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 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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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

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...

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...

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...

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...

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...

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...

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...

differential field extension generated by the solutions of a linear differential equation, using the differential Galois group of the field extension. A major...

In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q {\displaystyle...