Algebraic extension information

In 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...

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, 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...

and 1/s are all algebraic. An algebraic extension L / K {\displaystyle L/K} is an extension such that every element of L is algebraic over K. Equivalently...

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

degree is nonzero. Transcendental extensions are widely used in algebraic geometry. For example, the dimension of an algebraic variety is the transcendence...

F is algebraically closed if and only if it has no proper finite extension because if, within the previous proof, the term "algebraic extension" is replaced...

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...

of Algebraic Schemes. Springer Science & Business Media. ISBN 978-3-540-30615-3. algebra extension at nLab infinitesimal extension at nLab Extension of...

algebras and their representation theory, a Lie algebra extension e is an enlargement of a given Lie algebra g by another Lie algebra h. Extensions arise...

for deciding if an algebraic equation may be solved in terms of radicals. In field theory, an algebraic extension is an extension such that every element...

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...

In Lie group theory, central extensions arise in connection with algebraic topology. Roughly speaking, central extensions of Lie groups by discrete groups...

irreducible polynomial over k is separable. Every finite extension of k is separable. Every algebraic extension of k is separable. Either k has characteristic 0...

In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root...

1 + i {\displaystyle 1+i} is algebraic because it is a root of x4 + 4. All integers and rational numbers are algebraic, as are all roots of integers...

general notion of radical extensions. An algebraic extension E ⊇ F {\displaystyle E\supseteq F} is a purely inseparable extension if and only if for every...

mathematics, if L is an extension field of K, then an element a of L is called an algebraic element over K, or just algebraic over K, if there exists...

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

In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in...

Look up algebraic in Wiktionary, the free dictionary. Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic...

factors. Algebraic closure An algebraic closure of a field F is an algebraic extension of F which is algebraically closed. Every field has an algebraic closure...

are also algebraically independent over Q {\displaystyle \mathbb {Q} } . Given a field extension L / K {\displaystyle L/K} that is not algebraic, Zorn's...

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

The theorem states that given any field F {\displaystyle F} , an algebraic extension field E {\displaystyle E} of F {\displaystyle F} and an isomorphism...

theory Linear extension, in order theory Sheaf extension, in algebraic geometry Tietze extension theorem, in topology Whitney extension theorem, in differential...

theory) Field extension Algebraic extension Splitting field Algebraically closed field Algebraic element Algebraic closure Separable extension Separable polynomial...

Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems...