Field extension: Difference between revisions

In [[mathematics]], particularly in [[algebra]], a '''field extension''' (denoted <math>L/K</math>) is a pair of [[Field (mathematics)|fields]] <math>K \subseteq L,</math>, such that the operations of ''K'' are those of ''L'' [[Restriction (mathematics)|restricted]] to ''K''. In this case, ''L'' is an '''extension field''' of ''K'' and ''K'' is a '''subfield''' of ''L''.<ref>{{harvtxt|Fraleigh|1976|p=293}}</ref><ref>{{harvtxt|Herstein|1964|p=167}}</ref><ref>{{harvtxt|McCoy|1968|p=116}}</ref> For example, under the usual notions of [[addition]] and [[multiplication]], the [[complex number]]s are an extension field of the [[real number]]s; the real numbers are a subfield of the complex numbers.

If ''K'' is a subfield of ''L'', then ''L'' is an '''extension field''' or simply '''extension''' of ''K'', and this pair of fields is a '''field extension'''. Such a field extension is denoted ''<math>L'' / ''K''</math> (read as "''L'' over ''K''").

If ''L'' is an extension of ''F'', which is in turn an extension of ''K'', then ''F'' is said to be an '''intermediate field''' (or '''intermediate extension''' or '''subextension''') of ''<math>L'' / ''K''</math>.

Given a field extension {{nowrap|''<math>L'' / ''K''}}</math>, the larger field ''L'' is a ''K''-[[vector space]]. The [[dimension (vector space)|dimension]] of this vector space is called the [[degree of a field extension|'''degree''' of the extension]] and is denoted by <math>[''L''&nbsp;:&nbsp;''K'']</math>.

The degree of an extension is 1 if and only if the two fields are equal. In this case, the extension is a '''{{vanchor|trivial extension}}'''. Extensions of degree 2 and 3 are called '''quadratic extensions''' and '''cubic extensions''', respectively. A '''finite extension''' is an extension that has a finite degree.

Given two extensions {{nowrap|''<math>L'' / ''K''}}</math> and {{nowrap|''<math>M'' / ''L''}}</math>, the extension {{nowrap|''<math>M'' / ''K''}}</math> is finite if and only if both {{nowrap|''<math>L'' / ''K''}}</math> and {{nowrap|''<math>M'' / ''L''}}</math> are finite. In this case, one has

Given a field extension ''<math>L'' / ''K''</math> and a subset ''S'' of ''L'', there is a smallest subfield of ''L'' that contains ''K'' and ''S''. It is the intersection of all subfields of ''L'' that contain ''K'' and ''S'', and is denoted by ''K''(''S'') (read as "''K'' ''{{vanchor|adjoin}}'' ''S''"). One says that ''K''(''S'') is the field ''generated'' by ''S'' over ''K'', and that ''S'' is a [[generating set]] of ''K''(''S'') over ''K''. When <math>S=\{x_1, \ldots, x_n\}</math> is finite, one writes <math>K(x_1, \ldots, x_n)</math> instead of <math>K(\{x_1, \ldots, x_n\}),</math> and one says that ''K''(''S'') is {{vanchor|finitely generated}} over ''K''. If ''S'' consists of a single element ''s'', the extension {{nowrap|''K''(''s'') / ''K''}} is called a [[simple extension]]<ref>{{harvtxt|Fraleigh|1976|p=298}}</ref><ref>{{harvtxt|Herstein|1964|p=193}}</ref> and ''s'' is called a [[primitive element (field theory)|primitive element]] of the extension.<ref>{{harvtxt|Fraleigh|1976|p=363}}</ref>

An element ''x'' of a field extension {{nowrap|''<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 ''algebraic extension'' {{nowrap|''<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>

Given a field extension {{nowrap|''<math>L'' / ''K''}}</math>, a subset ''S'' of ''L'' is called [[algebraically independent]] over ''K'' if no non-trivial polynomial relation with coefficients in ''K'' exists among the elements of ''S''. The largest cardinality of an algebraically independent set is called the [[transcendence degree]] of ''L''/''K''. It is always possible to find a set ''S'', algebraically independent over ''K'', such that ''L''/''K''(''S'') is algebraic. Such a set ''S'' is called a [[transcendence basis]] of ''L''/''K''. All transcendence bases have the same cardinality, equal to the transcendence degree of the extension. An extension ''<math>L''/''K''</math> is said to be '''{{visible anchor|purely transcendental}}''' if and only if there exists a transcendence basis ''S'' of ''<math>L''/''K''</math> such that ''L'' = ''K''(''S''). Such an extension has the property that all elements of ''L'' except those of ''K'' are transcendental over ''K'', but, however, there are extensions with this property which are not purely transcendental—a class of such extensions take the form ''L''/''K'' where both ''L'' and ''K'' are algebraically closed. In addition, if ''L''/''K'' is purely transcendental and ''S'' is a transcendence basis of the extension, it doesn't necessarily follow that ''L'' = ''K''(''S'').

An algebraic extension ''<math>L''/''K''</math> is called [[normal extension|normal]] if every [[irreducible polynomial]] in ''K''[''X''] that has a root in ''L'' completely factors into linear factors over ''L''. Every algebraic extension ''F''/''K'' admits a normal closure ''L'', which is an extension field of ''F'' such that ''<math>L''/''K''</math> is normal and which is minimal with this property.

An algebraic extension ''<math>L''/''K''</math> is called [[separable extension|separable]] if the minimal polynomial of every element of ''L'' over ''K'' is [[separable polynomial|separable]], i.e., has no repeated roots in an algebraic closure over ''K''. A [[Galois extension]] is a field extension that is both normal and separable.

Given any field extension ''<math>L''/''K''</math>, we can consider its '''automorphism group''' <math>\text{Aut}(''L''/''K'')</math>, consisting of all field [[automorphism]]s ''α'': ''L'' → ''L'' with ''α''(''x'') = ''x'' for all ''x'' in ''K''. When the extension is Galois this automorphism group is called the [[Galois group]] of the extension. Extensions whose Galois group is [[abelian group|abelian]] are called [[abelian extension]]s.

For a given field extension ''<math>L''/''K''</math>, one is often interested in the intermediate fields ''F'' (subfields of ''L'' that contain ''K''). The significance of Galois extensions and Galois groups is that they allow a complete description of the intermediate fields: there is a [[bijection]] between the intermediate fields and the [[subgroup]]s of the Galois group, described by the [[fundamental theorem of Galois theory]].