Field extension: Difference between revisions
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In [[mathematics]], a '''field extension''' of a [[field (mathematics)|field]] ''F'' is a field ''E'' such that ''F'' is a [[subfield]] of ''E''. We say that ''E''/''F'' is an extension, or that ''E'' is an extension field of ''F''. | |||
For example, the field of [[complex number]]s '''C''' is an extension of the field of [[real number]]s '''R'''. | |||
If ''E''/''F'' is an extension then ''E'' is a [[vector space]] over ''F''. The ''degree'' or ''index'' of the field extension [''E'':''F''] is the [[dimension]] of ''E'' as an ''F''-vector space. The extension '''C'''/'''R''' has degree 2. An extension of degree 2 is ''quadratic''. | If ''E''/''F'' is an extension then ''E'' is a [[vector space]] over ''F''. The ''degree'' or ''index'' of the field extension [''E'':''F''] is the [[dimension]] of ''E'' as an ''F''-vector space. The extension '''C'''/'''R''' has degree 2. An extension of degree 2 is ''quadratic''. | ||
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:<math>[K:F] = [K:E] \cdot [E:F] \,</math> | :<math>[K:F] = [K:E] \cdot [E:F] \,</math> | ||
An element of an extension field ''E''/''F'' is ''algebraic'' over ''F'' if it satisfies a [[polynomial]] with coefficients in ''F''. An extension is ''algebraic'' if every element of ''E'' is algebraic over ''F''. An extension of finite degree is algebraic, but the converse need not hold. For example, the field of all [[algebraic number]]s over '''Q''' is an algebraic extension but not of finite degree. | ==Algebraic extension== | ||
An element of an extension field ''E''/''F'' is ''algebraic'' over ''F'' if it satisfies a [[polynomial]] with coefficients in ''F'', and ''transcendental'' over ''F'' if it is not algebraic. An extension is ''algebraic'' if every element of ''E'' is algebraic over ''F''. An extension of finite degree is algebraic, but the converse need not hold. For example, the field of all [[algebraic number]]s over '''Q''' is an algebraic extension but not of finite degree. | |||
==Separable extension== | |||
An element of an extension field is ''separable'' over ''F'' if it is algebraic and its [[minimal polynomial]] over ''F'' has distinct roots. Every algebraic element is separable over a field of [[Characteristic of a field|characteristic]] zero. An extension is ''separable'' if all its elements are. A field is ''perfect'' if all finite degree extensions are separable. For example, a [[finite field]] is perfect. | |||
==Simple extension== | |||
A '''simple extension''' is one which is generated by a single element, say ''a'', and a generating element is a '''primitive element'''. The extension ''F''(''a'') is formed by the polynomial [[ring]] ''F''[''a''] if ''a'' is algebraic, otherwise it is the [[rational function]] field ''F''(''a''). | |||
The '''theorem of the primitive element''' states that a finite degree extension ''E''/''F'' is simple if and only if there are only finitely many intermediate fields between ''E'' and ''F''; as a consequence, every finite degree separable extension is simple. | |||
==References== | ==References== | ||
* {{cite book | author=A.G. Howson | title=A handbook of terms used in algebra and analysis | publisher=[[Cambridge University Press]] | year=1972 | isbn=0-521-09695-2 | pages=72-73 }} | * {{cite book | author=A.G. Howson | title=A handbook of terms used in algebra and analysis | publisher=[[Cambridge University Press]] | year=1972 | isbn=0-521-09695-2 | pages=72-73 }} | ||
* {{ cite book | author=P.J. McCarthy | title=Algebraic extensions of fields | publisher=[[Dover Publications]] | year=1991 | isbn=0-486-66651-4 }} | |||
* {{cite book | author=I.N. Stewart | authorlink=Ian Stewart (mathematician) | title=Galois theory | publisher=Chapman and Hall | year=1973 | isbn=0-412-10800-3 | pages=33-48 }}[[Category:Suggestion Bot Tag]] | |||
Latest revision as of 06:01, 16 August 2024
In mathematics, a field extension of a field F is a field E such that F is a subfield of E. We say that E/F is an extension, or that E is an extension field of F.
For example, the field of complex numbers C is an extension of the field of real numbers R.
If E/F is an extension then E is a vector space over F. The degree or index of the field extension [E:F] is the dimension of E as an F-vector space. The extension C/R has degree 2. An extension of degree 2 is quadratic.
The tower law for extensions states that if K/E is another extension, then
![{\displaystyle [K:F]=[K:E]\cdot [E:F]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c8a9f48db475e81b37406a83720a37f3084b508)
Algebraic extension
An element of an extension field E/F is algebraic over F if it satisfies a polynomial with coefficients in F, and transcendental over F if it is not algebraic. An extension is algebraic if every element of E is algebraic over F. An extension of finite degree is algebraic, but the converse need not hold. For example, the field of all algebraic numbers over Q is an algebraic extension but not of finite degree.
Separable extension
An element of an extension field is separable over F if it is algebraic and its minimal polynomial over F has distinct roots. Every algebraic element is separable over a field of characteristic zero. An extension is separable if all its elements are. A field is perfect if all finite degree extensions are separable. For example, a finite field is perfect.
Simple extension
A simple extension is one which is generated by a single element, say a, and a generating element is a primitive element. The extension F(a) is formed by the polynomial ring F[a] if a is algebraic, otherwise it is the rational function field F(a).
The theorem of the primitive element states that a finite degree extension E/F is simple if and only if there are only finitely many intermediate fields between E and F; as a consequence, every finite degree separable extension is simple.
References
- A.G. Howson (1972). A handbook of terms used in algebra and analysis. Cambridge University Press, 72-73. ISBN 0-521-09695-2.
- P.J. McCarthy (1991). Algebraic extensions of fields. Dover Publications. ISBN 0-486-66651-4.
- I.N. Stewart (1973). Galois theory. Chapman and Hall, 33-48. ISBN 0-412-10800-3.