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.

See also: Algebraic extension, [Galois theory]?