Editing Field extension (section)

== Algebraic extension ==
{{main|Algebraic extension|Algebraic element}}
An element ''x'' of a field extension <math>L/K</math> is algebraic over ''K'' if it is a [[root of a function|root]] of a nonzero [[polynomial]] with coefficients in ''K''. For example, <math>\sqrt 2</math> is algebraic over the rational numbers, because it is a root of <math>x^2-2.</math> If an element ''x'' of ''L'' is algebraic over ''K'', the [[monic polynomial]] of lowest degree that has ''x'' as a root is called the [[minimal polynomial (field theory)|minimal polynomial]] of ''x''. This minimal polynomial is [[irreducible polynomial|irreducible]] over ''K''.

An element ''s'' of ''L'' is algebraic over ''K'' if and only if the simple extension {{nowrap|''K''(''s'') /''K''}} is a finite extension. In this case the degree of the extension equals the degree of the minimal polynomial, and a basis of the ''K''-[[vector space]] ''K''(''s'') consists of <math>1, s, s^2, \ldots, s^{d-1},</math> where ''d'' is the degree of the minimal polynomial.

The set of the elements of ''L'' that are algebraic over ''K'' form a subextension, which is called the [[algebraic closure]] of ''K'' in ''L''. This results from the preceding characterization: if ''s'' and ''t'' are algebraic, the extensions {{nowrap|''K''(''s'') /''K''}} and {{nowrap|''K''(''s'')(''t'') /''K''(''s'')}} are finite. Thus {{nowrap|''K''(''s'', ''t'') /''K''}} is also finite, as well as the sub extensions {{nowrap|''K''(''s'' ± ''t'') /''K''}}, {{nowrap|''K''(''st'') /''K''}} and {{nowrap|''K''(1/''s'') /''K''}} (if {{nowrap|''s'' ≠ 0}}). It follows that {{nowrap|''s'' ± ''t''}}, ''st'' and 1/''s'' are all algebraic.

An ''algebraic extension'' <math>L/K</math> is an extension such that every element of ''L'' is algebraic over ''K''. Equivalently, an algebraic extension is an extension that is generated by algebraic elements. For example, <math>\Q(\sqrt 2, \sqrt 3)</math> is an algebraic extension of <math>\Q</math>, because <math>\sqrt 2</math> and <math>\sqrt 3</math> are algebraic over <math>\Q.</math>

A simple extension is algebraic [[if and only if]] it is finite. This implies that an extension is algebraic if and only if it is the union of its finite subextensions, and that every finite extension is algebraic. 

Every field ''K'' has an algebraic closure, which is [[up to]] an isomorphism the largest extension field of ''K'' which is algebraic over ''K'', and also the smallest extension field such that every polynomial with coefficients in ''K'' has a root in it. For example, <math>\Complex</math> is an algebraic closure of <math>\R</math>, but not an algebraic closure of <math>\Q</math>, as it is not algebraic over <math>\Q</math> (for example {{pi}} is not algebraic over <math>\Q</math>).