Field extension: Difference between revisions
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==Simple extension== | ==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''). | 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 }} | ||
Revision as of 07:19, 20 December 2008
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.
Foe 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)
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.