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 is, every element of L is a root of a non-zero polynomial with coefficients in K.[1][2] A field extension that is not algebraic, is said to be transcendental, and must contain transcendental elements, that is, elements that are not algebraic.
All transcendental extensions are of infinite degree.
This in turn implies that all finite extensions are algebraic.
[5] The converse is not true however: there are infinite extensions which are algebraic.
[7] Let E be an extension field of K, and a ∈ E. The smallest subfield of E that contains K and a is commonly denoted
If a is algebraic over K, then the elements of K(a) can be expressed as polynomials in a with coefficients in K; that is,
[11] An extension L/K is algebraic if and only if every sub K-algebra of L is a field.
The following three properties hold:[12] These finitary results can be generalized using transfinite induction: This fact, together with Zorn's lemma (applied to an appropriately chosen poset), establishes the existence of algebraic closures.
Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding of M into N is called an algebraic extension if for every x in N there is a formula p with parameters in M, such that p(x) is true and the set is finite.
Given a field k and a field K containing k, one defines the relative algebraic closure of k in K to be the subfield of K consisting of all elements of K that are algebraic over k, that is all elements of K that are a root of some nonzero polynomial with coefficients in k.