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


 * $$[K:F] = [K:E] \cdot [E:F] \,$$

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.