9.26 Transcendence
We recall the standard definitions.
Definition 9.26.1. Let K/k be a field extension.
A collection of elements \{ x_ i\} _{i \in I} of K is called algebraically independent over k if the map
k[X_ i; i\in I] \longrightarrow K
which maps X_ i to x_ i is injective.
The field of fractions of a polynomial ring k[x_ i; i \in I] is denoted k(x_ i; i\in I).
A purely transcendental extension of k is any field extension K/k isomorphic to the field of fractions of a polynomial ring over k.
A transcendence basis of K/k is a collection of elements \{ x_ i\} _{i \in I} which are algebraically independent over k and such that the extension K/k(x_ i; i\in I) is algebraic.
Example 9.26.2. The field \mathbf{Q}(\pi ) is purely transcendental because \pi isn't the root of a nonzero polynomial with rational coefficients. In particular, \mathbf{Q}(\pi ) \cong \mathbf{Q}(x).
Lemma 9.26.3. Let E/F be a field extension. A transcendence basis of E over F exists. Any two transcendence bases have the same cardinality.
Proof.
Let A be an algebraically independent subset of E. Let G be a subset of E containing A that generates E/F. We claim we can find a transcendence basis B such that A \subset B \subset G. To prove this, consider the collection \mathcal{B} of algebraically independent subsets whose members are subsets of G that contain A. Define a partial ordering on \mathcal{B} using inclusion. Then \mathcal{B} contains at least one element A. The union of the elements of a totally ordered subset T of \mathcal{B} is an algebraically independent subset of E over F since any algebraic dependence relation would have occurred in one of the elements of T (since polynomials only involve finitely many variables). The union also contains A and is contained in G. By Zorn's lemma, there is a maximal element B \in \mathcal{B}. Now we claim E is algebraic over F(B). This is because if it wasn't then there would be an element f \in G transcendental over F(B) since F(G) = E. Then B \cup \{ f\} would be algebraically independent contradicting the maximality of B. Thus B is our transcendence basis.
Let B and B' be two transcendence bases. Without loss of generality, we can assume that |B'| \leq |B|. Now we divide the proof into two cases: the first case is that B is an infinite set. Then for each \alpha \in B', there is a finite set B_{\alpha } \subset B such that \alpha is algebraic over F(B_{\alpha }) since any algebraic dependence relation only uses finitely many indeterminates. Then we define B^* = \bigcup _{\alpha \in B'} B_{\alpha }. By construction, B^* \subset B, but we claim that in fact the two sets are equal. To see this, suppose that they are not equal, say there is an element \beta \in B \setminus B^*. We know \beta is algebraic over F(B') which is algebraic over F(B^*). Therefore \beta is algebraic over F(B^*), a contradiction. So |B| \leq |\bigcup _{\alpha \in B'} B_{\alpha }|. Now if B' is finite, then so is B so we can assume B' is infinite; this means
|B| \leq |\bigcup \nolimits _{\alpha \in B'} B_{\alpha }| = |B'|
because each B_\alpha is finite and B' is infinite. Therefore in the infinite case, |B| = |B'|.
Now we need to look at the case where B is finite. In this case, B' is also finite, so suppose B = \{ \alpha _1, \ldots , \alpha _ n\} and B' = \{ \beta _1, \ldots , \beta _ m\} with m \leq n. We perform induction on m: if m = 0 then E/F is algebraic so B = \emptyset so n = 0. If m > 0, there is an irreducible polynomial f \in F[x, y_1, \ldots , y_ n] such that f(\beta _1, \alpha _1, \ldots , \alpha _ n) = 0 and such that x occurs in f. Since \beta _1 is not algebraic over F, f must involve some y_ i so without loss of generality, assume f uses y_1. Let B^* = \{ \beta _1, \alpha _2, \ldots , \alpha _ n\} . We claim that B^* is a basis for E/F. To prove this claim, we see that we have a tower of algebraic extensions
E/ F(B^*, \alpha _1) / F(B^*)
since \alpha _1 is algebraic over F(B^*). Now we claim that B^* (counting multiplicity of elements) is algebraically independent over F because if it weren't, then there would be an irreducible g\in F[x, y_2, \ldots , y_ n] such that g(\beta _1, \alpha _2, \ldots , \alpha _ n) = 0 which must involve x making \beta _1 algebraic over F(\alpha _2, \ldots , \alpha _ n) which would make \alpha _1 algebraic over F(\alpha _2, \ldots , \alpha _ n) which is impossible. So this means that \{ \alpha _2, \ldots , \alpha _ n\} and \{ \beta _2, \ldots , \beta _ m\} are bases for E over F(\beta _1) which means by induction, m = n.
\square
Definition 9.26.4. Let K/k be a field extension. The transcendence degree of K over k is the cardinality of a transcendence basis of K over k. It is denoted \text{trdeg}_ k(K).
Lemma 9.26.5. Let L/K/k be field extensions. Then
\text{trdeg}_ k(L) = \text{trdeg}_ K(L) + \text{trdeg}_ k(K).
Proof.
Choose a transcendence basis A \subset K of K over k. Choose a transcendence basis B \subset L of L over K. Then it is straightforward to see that A \cup B is a transcendence basis of L over k.
\square
Example 9.26.6. Consider the field extension \mathbf{Q}(e, \pi ) formed by adjoining the numbers e and \pi . This field extension has transcendence degree at least 1 since both e and \pi are transcendental over the rationals. However, this field extension might have transcendence degree 2 if e and \pi are algebraically independent. Whether or not this is true is unknown and whence the problem of determining \text{trdeg}(\mathbf{Q}(e, \pi )) is open.
Example 9.26.7. Let F be a field and E = F(t). Then \{ t\} is a transcendence basis since E = F(t). However, \{ t^2\} is also a transcendence basis since F(t)/F(t^2) is algebraic. This illustrates that while we can always decompose an extension E/F into an algebraic extension E/F' and a purely transcendental extension F'/F, this decomposition is not unique and depends on choice of transcendence basis.
Example 9.26.8. Let X be a compact Riemann surface. Then the function field \mathbf{C}(X) (see Example 9.3.6) has transcendence degree one over \mathbf{C}. In fact, any finitely generated extension of \mathbf{C} of transcendence degree one arises from a Riemann surface. There is even an equivalence of categories between the category of compact Riemann surfaces and (non-constant) holomorphic maps and the opposite of the category of finitely generated extensions of \mathbf{C} of transcendence degree 1 and morphisms of \mathbf{C}-algebras. See [Forster].
There is an algebraic version of the above statement as well. Given an (irreducible) algebraic curve in projective space over an algebraically closed field k (e.g. the complex numbers), one can consider its “field of rational functions”: basically, functions that look like quotients of polynomials, where the denominator does not identically vanish on the curve. There is a similar anti-equivalence of categories (Algebraic Curves, Theorem 53.2.6) between smooth projective curves and non-constant morphisms of curves and finitely generated extensions of k of transcendence degree one. See [H].
Definition 9.26.9. Let K/k be a field extension.
The algebraic closure of k in K is the subfield k' of K consisting of elements of K which are algebraic over k.
We say k is algebraically closed in K if every element of K which is algebraic over k is contained in k.
Lemma 9.26.10. Let k'/k be a finite extension of fields. Let k'(x_1, \ldots , x_ r)/k(x_1, \ldots , x_ r) be the induced extension of purely transcendental extensions. Then [k'(x_1, \ldots , x_ r) : k(x_1, \ldots , x_ r)] = [k' : k] < \infty .
Proof.
By multiplicativity of degrees of extensions (Lemma 9.7.7) it suffices to prove this when k' is generated by a single element \alpha \in k' over k. Let f \in k[T] be the minimal polynomial of \alpha over k. Then k'(x_1, \ldots , x_ r) is generated by \alpha , x_1, \ldots , x_ r over k and hence k'(x_1, \ldots , x_ r) is generated by \alpha over k(x_1, \ldots , x_ r). Thus it suffices to show that f is still irreducible as an element of k(x_1, \ldots , x_ r)[T]. We only sketch the proof. It is clear that f is irreducible as an element of k[x_1, \ldots , x_ r, T] for example because f is monic as a polynomial in T and any putative factorization in k[x_1, \ldots , x_ r, T] would lead to a factorization in k[T] by setting x_ i equal to 0. By Gauss' lemma we conclude.
\square
Lemma 9.26.11. Let K/k be a finitely generated field extension. The algebraic closure of k in K is finite over k.
Proof.
Let x_1, \ldots , x_ r \in K be a transcendence basis for K over k. Then n = [K : k(x_1, \ldots , x_ r)] < \infty . Suppose that k \subset k' \subset K with k'/k finite. In this case [k'(x_1, \ldots , x_ r) : k(x_1, \ldots , x_ r)] = [k' : k] < \infty , see Lemma 9.26.10. Hence
[k' : k] = [k'(x_1, \ldots , x_ r) : k(x_1, \ldots , x_ r)] \leq [K : k(x_1, \ldots , x_ r)] = n.
In other words, the degrees of finite subextensions are bounded and the lemma follows.
\square
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