; that is, an element that is not a root of any univariate polynomial with coefficients in
Transcendental extensions are widely used in algebraic geometry.
For example, the dimension of an algebraic variety is the transcendence degree of its function field.
Also, global function fields are transcendental extensions of degree one of a finite field, and play in number theory in positive characteristic a role that is very similar to the role of algebraic number fields in characteristic zero.
Zorn's lemma shows there exists a maximal linearly independent subset of a vector space (i.e., a basis).
A similar argument with Zorn's lemma shows that, given a field extension L / K, there exists a maximal algebraically independent subset of L over K.[1] It is then called a transcendence basis.
By maximality, an algebraically independent subset S of L over K is a transcendence basis if and only if L is an algebraic extension of K(S), the field obtained by adjoining the elements of S to K. The exchange lemma (a version for algebraically independent sets[2]) implies that if S and S' are transcendence bases, then S and S' have the same cardinality.
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 finitary matroids (pregeometries).
Any finitary matroid has a basis, and all bases have the same cardinality.
[3] If G is a generating set of L (i.e., L = K(G)), then a transcendence basis for L can be taken as a subset of G. Thus,
the minimum cardinality of generating sets of L over K. In particular, a finitely generated field extension admits a finite transcendence basis.
[5] Over a perfect field, every finitely generated field extension is separably generated; i.e., it admits a finite separating transcendence basis.
[6] If M / L and L / K are field extensions, then 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. If the set S is algebraically independent over K, then the field K(S) is isomorphic to the field of rational functions over K in a set of variables of the same cardinality as S. Each such rational function is a fraction of two polynomials in finitely many of those variables, with coefficients in K. Two algebraically closed fields are isomorphic if and only if they have the same characteristic and the same transcendence degree over their prime field.
denote the fields of fractions of A and B, then the transcendence degree of B over A is defined as the transcendence degree of the field extension
The Noether normalization lemma implies that if R is an integral domain that is a finitely generated algebra over a field k, then the Krull dimension of R is the transcendence degree of R over k. This has the following geometric interpretation: if X is an affine algebraic variety over a field k, the Krull dimension of its coordinate ring equals the transcendence degree of its function field, and this defines the dimension of X.
It follows that, if X is not an affine variety, its dimension (defined as the transcendence degree of its function field) can also be defined locally as the Krull dimension of the coordinate ring of the restriction of the variety to an open affine subset.
Transcendence bases are useful for proving various existence statements about field homomorphisms.
The field L is the algebraic closure of K(S) and algebraic closures are unique up to isomorphism; this means that the automorphism can be further extended from K(S) to L. As another application, we show that there are (many) proper subfields of the complex number field C which are (as fields) isomorphic to C. For the proof, take a transcendence basis S of C / Q.
S is an infinite (even uncountable) set, so there exist (many) maps f: S → S which are injective but not surjective.
Any such map can be extended to a field homomorphism Q(S) → Q(S) which is 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(X) denotes the field of (globally defined) meromorphic functions on it, then trdegC(K(X)) ≤ n.