Field extension information

mathematics, particularly in algebra, a field extension (denoted L / K {\displaystyle L/K} ) is a pair of fields K ⊆ L {\displaystyle K\subseteq L} , such...

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...

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...

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...

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, 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...

a larger domain Extension of a polyhedron, in geometry Exterior algebra, Grassmann's theory of extension, in geometry Field extension, in Galois theory...

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 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...

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...

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...

In mathematics and more specifically in field theory, a radical extension of a field K is an extension of K that is obtained by adjoining a sequence of...

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...

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

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...

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...

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...

equation xpn − x = 0. Any finite field extension of a finite field is separable and simple. That is, if E is a finite field and F is a subfield of E, then...

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

appear naturally in the study of polynomial factorization and algebraic field extensions. It is helpful to compare irreducible polynomials to prime numbers:...

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...

Given a finite field extension K / F {\displaystyle K/F} over a locally compact field F {\displaystyle F} , there is at most one unique field norm | ⋅ | K...

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...