Algebraic extension

Field extension via adjoining solutions to polynomials with coefficients in the subfield

In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K.[1][2] Field extensions that are not algebraic, i.e. which contain transcendental elements, are called transcendental.[3][4]

For example, the field extension R/Q, that is the field of real numbers as an extension of the field of rational numbers, is transcendental,[5] while the field extensions C/R[6] and Q(2)/Q[7] are algebraic, where C is the field of complex numbers.

All transcendental extensions are of infinite degree. This in turn implies that all finite extensions are algebraic.[8] The converse is not true however: there are infinite extensions which are algebraic.[9] For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers.[10]

Let E be an extension field of K, and a ∈ E. If a is algebraic over K, then K(a), the set of all polynomials in a with coefficients in K, is not only a ring but a field: K(a) is an algebraic extension of K which has finite degree over K.[11] The converse is not true. Q[π] and Q[e] are fields but π and e are transcendental over Q.[12]

An algebraically closed field F has no proper algebraic extensions, that is, no algebraic extensions E with F < E.[13] An example is the field of complex numbers. 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.[14]

An extension L/K is algebraic if and only if every sub K-algebra of L is a field.

The class of algebraic extensions forms a distinguished class of field extensions, that is, the following three properties hold:[15]

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. 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.