Editing Field extension (section)

==Extension field==

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

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

Given a field extension <math>L/K</math>, the larger field <math>L</math> is a <math>K</math>-[[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: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 <math>L/K</math> and <math>M/L</math>, the extension <math>M/K</math> is finite if and only if both <math>L/K</math> and <math>M/L</math> are finite. In this case, one has

:<math>[M : K]=[M : L]\cdot[L : K].</math>

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

An extension field of the form <math>K(S)</math> is often said to result from the ''{{vanchor|adjunction}}'' of <math>S</math> to <math>K</math>.<ref>{{harvtxt|Fraleigh|1976|p=319}}</ref><ref>{{harvtxt|Herstein|1964|p=169}}</ref>

In [[characteristic of a ring|characteristic]] 0, every finite extension is a simple extension. This is the [[primitive element theorem]], which does not hold true for fields of non-zero characteristic.

If a simple extension <math>K(s)/K</math> is not finite, the field <math>K(s)</math> is isomorphic to the field of [[rational fraction]]s in <math>s</math> over <math>K</math>.