# Field extension

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

## 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.