In [[abstract algebra]], a [[field extension]] ''L'' /''K'' is called '''algebraic''' if every element of ''L'' is [[algebraic element|algebraic]] over ''K'', i.e. if every element of ''L'' is a [[root (mathematics)|root]] of some non-zero [[polynomial]] with coefficients in ''K''. Field extensions which are not algebraic, i.e. which contain [[transcendental element]]s, are called '''transcendental'''.

For example, the field extension '''R'''/'''Q''', that is the field of [[real number]]s as an extension of the field of [[rational number]]s, is transcendental, while the field extensions '''C'''/'''R''' and '''Q'''(√2)/'''Q''' are algebraic, where '''C''' is the field of [[complex number]]s.

All transcendental extensions are of [[degree of a field extension|infinite degree]]. This in turn implies that all finite extensions are algebraic. The converse is not true however: there are infinite extensions which are algebraic. For instance, the field of all [[algebraic number]]s is an infinite algebraic extension of the rational numbers.

If ''a'' is algebraic over ''K'', then ''K''[''a''], the set of all polynomials in ''a'' with coefficients in ''K'', is a field. It is an algebraic field extension of ''K'' which has finite degree over ''K''. In the special case where ''K''='''Q''' is the [[rational number|field of rational numbers]], '''Q'''[''a''] is an example of an [[algebraic number field]].

A field with no proper algebraic extensions is called [[algebraically closed field|algebraically closed]]. An example is the field of [[complex number]]s. Every field has an algebraic extension which is algebraically closed (called its [[algebraic closure]]), but proving this in general requires some form of the [[axiom of choice]].

An extension ''L''/''K'' is algebraic [[if and only if]] every sub ''K''-algebra of ''L'' is a [[field (mathematics)|field]].

==Generalizations== {{main article|extension (model theory)}}

[[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

:{''y'' in ''N'' | ''p''(''y'')}

is finite. It turns out that applying this definition to the theory of fields gives the usual definition of algebraic extension. The [[Galois group]] of ''N'' over ''M'' can again be defined as the group of automorphisms, and it turns out that most of the theory of Galois groups can be developed for the general case.

== See also ==

* [[Algebraic number]]
* [[Algebraically closed field]]
* [[Algebraic closure]]

[[Category:Abstract algebra]]
[[Category:Algebra]]

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