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''. | 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''. | ||
Revision as of 05:36, 5 February 2009
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.