Transcendental extension

In mathematics, a transcendental extension L / K is a field extension such that there exists a transcendental element in L over K; i.e., an element that is not a root of any polynomial over K. In other words, a transcendental extension is a field extension that is not algebraic. For example, $\mathbb {C} ,\mathbb {R}$ are both transcendental extensions over $\mathbb {Q}$.

The cardinality of a maximal algebraically independent subset of L over K, like a basis for a vector space, turns out to be independent of a choice of the set and the cardinality is called transcendence degree. Thus, a field extension is transcendental extension if and only if the transcendence degree is positive.

Transcendental extensions appear especially in algebraic geometry: if $f:X\to Y$ is a morphism between algebraic varieties that is dominant (has dense image), then the pull-back of rational functions determine the field extension $k(X)/f^{*}(k(Y))$. Then the transcendence degree is the relative dimension of the morphism.

Transcendence basis

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. It is then called a transcendence basis. Note, by maximality, if S is a transcendence basis, then L is an algebraic extension of the field K(S) (the field obtained by adjoining the elements of S to K).

The exchange lemma (a version for algebraically independent sets) implies that if S, S' are transcendence bases, then S and S' have the same cardinality (i.e., there exists a bijection between them). Then the common cardinality of transcendence bases is called the transcendence degree of L over K. There is thus an analogy: a transcendence basis and transcendence degree, on the one hand, and a basis and dimension on the other hand. 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.

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., the rational numbers field Q if L is of characteristic 0 and the finite field F_p 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(S).

A separating transcendence basis of L / K is a transcendence basis S such that L is a separable algebraic extension over K(S). Over a perfect field, every finitely generated field extension admits a finite separating transcendence basis.

Examples

• An extension is algebraic if and only if its transcendence degree is 0; the empty set serves as a transcendence basis here.
• The field of rational functions in n variables K(x₁,...,x_n) (i.e. the field of fractions of the polynomial ring K[x₁,...,x_n]) is a purely transcendental extension with transcendence degree n over K; we can for example take {x₁,...,x_n} as a transcendence base.
• More generally, the transcendence degree of the function field L of an n-dimensional algebraic variety over a ground field K is n.
• Q(√2, e) has transcendence degree 1 over Q because √2 is algebraic while e is transcendental.
• The transcendence degree of C or R over Q is the cardinality of the continuum. (Since Q is countable, the field Q(S) will have the same cardinality as S for any infinite set S, and any algebraic extension of Q(S) will have the same cardinality again.)
• The transcendence degree of Q(e, π) over Q is either 1 or 2; the precise answer is unknown because it is not known whether e and π are algebraically independent.
• If S is a compact Riemann surface, the field C(S) of meromorphic functions on S has transcendence degree 1 over C.

Facts

If M / L and L / K are field extensions, then

trdeg(M / K) = trdeg(M / L) + trdeg(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.

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.

Applications

Transcendence bases are a useful tool to prove various existence statements about field homomorphisms. Here is an example: Given an algebraically closed field L, a subfield K and a field automorphism f of K, there exists a field automorphism of L which extends f (i.e. whose restriction to K is f). For the proof, one starts with a transcendence basis S of L / K. The elements of K(S) are just quotients of polynomials in elements of S with coefficients in K; therefore the automorphism f can be extended to one of K(S) by sending every element of S to itself. 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. 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(X) denotes the field of (globally defined) meromorphic functions on it, then trdeg_C(K(X)) ≤ n.

See also

• Lüroth’s theorem, a theorem about purely transcendental extensions of degree one
• Dimension of an algebraic variety, which can be defined as a transcendence degree
• Regular extension
• Noether normalization lemma

References

• Milne, James, Field Theory (PDF)
• § 6.3. of Shimura, Goro (1971), Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Tokyo: Iwanami Shoten, Zbl 0221.10029