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.

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


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

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 numbers over Q is an algebraic extension but not of finite degree.

An element of E is transcendental over F if it is not algebraic.