Field extension

In abstract algebra, an extension of a field K is a field L such that K is a subfield of L. For example, C (the field of complex numbers) is an extension of R (the field of real numbers), and R is itself an extension of Q (the field of rational numbers). The notation L/K is sometimes used to denote the fact that L is a extension of K.

Given a field extension L/K, L can be considered as a vector space over K, with vector addition being the field addition on L, and scalar multiplication being a restriction of the field multiplication on L. The dimension of this vector space is called the degree of the extension, and is denoted [L : K]. The extension is said to be finite or infinite according as the degree is finite or infinite. For example, [C : R] = 2, so this extension is finite. By contrast, [R : Q] = c (the cardinality of the continuum), so this extension is infinite.

If L is an extension of K, then an element of L which is a root of a nonzero polynomial over K is said to be algebraic over K. If it is not algebraic then it is said to be transcendental. (The special case where L = C and K = Q is particularly important. See Algebraic number and Transcendental number.) If every element of L is algebraic over K, then the extension L/K is said to be algebraic, otherwise it is said to be transcendental. It can be shown that an extension is algebraic if and only if it is the union of its finite subextensions. In particular, every finite extension is algebraic. See Algebraic extension for more information on algebraic extensions.

See also: Galois group