In abstract algebra, the transcendence degree of a field extensionL /K is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independentsubset of L over K. A subset S of L is a transcendence basis of L /K if it is algebraically independent over K and if furthermore L is an algebraic extension of the fieldK. One can show that every field extension has a transcendence basis, and that all transcendence bases have the same cardinality; this cardinality is equal to the transcendence degree of the extension and is denoted trdegKL or trdeg. If no field K is specified, the transcendence degree of a field L is its degree relative to the prime field of the same characteristic, i.e., Q if L is of characteristic 0 and Fp if L is of characteristic p. The field extension L /K is purely transcendental if there is a subset S of L that is algebraically independent over K and such that L = K.
The transcendence degree of Q over Q is either 1 or 2; the precise answer is unknown because it is not known whether e and π are algebraically independent.
There is an analogy with the theory of vector space dimensions. The analogy matches algebraically independent sets with linearly independent sets; sets S such that L is algebraic over K with spanning sets; transcendence bases with bases; and transcendence degree with dimension. The fact that transcendence bases always exist requires the axiom of choice. The proof that any two bases have the same cardinality depends, in each setting, on an exchange lemma. This analogy can be made more formal, by observing that linear independence in vector spaces and algebraic independence in field extensions both form examples of matroids, called linear matroids and algebraic matroids respectively. Thus, the transcendence degree is the rank function of an algebraic matroid. Every linear matroid is isomorphic to an algebraic matroid, but not vice versa.
Facts
If M/L is a field extension and L /K is another field extension, then the transcendence degree of M/K is equal to the sum of the transcendence degrees of M/L and L/K. This is proven by showing that a transcendence basis of M/K can be obtained by taking the union of a transcendence basis of M/L and one of L /K.
Applications
Transcendence bases are a useful tool to prove various existence statements about field homomorphisms. Here is an example: Given an algebraically closed fieldL, a subfieldK and a field automorphismf of K, there exists a field automorphism of L which extends f. For the proof, one starts with a transcendence basis S of L/K. The elements of K are just quotients of polynomials in elements of S with coefficients in K; therefore the automorphism f can be extended to one of K by sending every element ofS to itself. The field L is the algebraic closure of K and algebraic closures are unique up to isomorphism; this means that the automorphism can be further extended from K to L. As another application, we show that there are proper subfields of the complex number fieldC which are isomorphic to C. For the proof, take a transcendence basis S of C/Q. S is an infinite set, so there exist maps f: S → S which are injective but not surjective. Any such map can be extended to a field homomorphism Q → Q which is not surjective. Such a field homomorphism can in turn be extended to the algebraic closure C, and the resulting field homomorphisms C → C are not surjective. The transcendence degree can give an intuitive understanding of the size of a field. For instance, a theorem due to Siegel states that if X is a compact, connected, complex manifold of dimension n and K denotes the field of meromorphic functions on it, then trdegC ≤ n.
