Field extension

In abstract algebra, an extension of a field K is a field L which contains K as a subfield. 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 often used to denote the fact that L is an extension of K.

More generally, an extension of K is a separate pair of fields K* and L, where L contains K* as a subfield, and K is isomorphic to K*. Where it does not cause confusion, we identify K and K*, as above and below.

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 M is an extension of L which is an extension of K, then [M : K] = [M : L].[L : K].

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. If every element of L \\ K is transcendental over K, then the extension is said to be pure 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. For example, C/R, being finite, is algebraic. But R/Q is transcendental, although not pure transcendental. See Algebraic extension for more information on algebraic extensions.

If L/K is a field extension and V is a subset of L, then the field K(V) is defined to be the smallest subfield of L which contains K and V. It consists of all those elements of L which can be gotten using a finite number of field operations +, -, *, / applied to elements from K and V. If L = K(V), we say that L is generated by V.

A field extension generated by a single element is called a simple extension. A simple extension is finite if generated by an algebraic element, and pure transcendental if generated by a transcendental element. For example, C is a simple extension of R, as it is generated by i (the square root of minus one). The extension R/Q is not simple, as it is neither finite nor pure transcendental.

A field extension which has a Galois group is called a Galois extension. If the Galois group is Abelian, then the extension is called an Abelian extension. For example, C/R is a Abelian extension, its Galois group being of order 2. But R/Q is not a Galois extension, as the only field automorphism of R is the identity automorphism.