An application of Zorn’s lemma: Transcendence bases

Zorn’s lemma is a very useful result when it comes to dealing with an infinite collection of things. In ZFC set theory it is equivalent to the well-ordering theorem (every set can be well-ordered) and to the axiom of choice (a Cartesian product of non-empty sets is non-empty). I happened to use it a few days ago in proving the existence of transcendence bases, hence this post!

Zorn’s lemma. If every chain in a poset P has an upper bound in P, then P contains a maximal element.

Let K/k be a field extension. We call a subset S\subseteq K algebraically independent if for any m\ge 0 and s_1,\dots,s_m\in S, p(s_1,\dots,s_m)=0 implies p=0, where p\in k[X_1,\dots,X_m] is a polynomial. A maximal (with respect to \subseteq) algebraically independent subset is called a transcendence base.

Theorem. Every field extension has a transcendence base.

Proof. If K/k is algebraic, the transcendence base is the empty set. Suppose that K/k is not algebraic. Let \mathcal F be the family of all algebraically independent subsets S\subseteq K. By Zorn’s lemma, it suffices to show that every chain in \mathcal F has an upper bound. Let \mathcal C be a chain in \mathcal F, and let

T=\displaystyle\bigcup_{S\in \mathcal C} S.

It suffices to show that T\in\mathcal F, since then T would be an upper bound for \mathcal C in \mathcal F.

Let P(m) be the statement:

If t_1,\dots,t_m\in T are distinct, and p(t_1,\dots, t_m)=0 for some p\in k[X_1,\dots,X_m], then p=0.

If t_1\in T, then t_1\in S for some S\in \mathcal C. Since \mathcal S is algebraically independent, p(t_1)=0 implies p=0 in k[X_1]. Thus P(1) is true.

Suppose P(m-1) is true. Let t_1,\dots,t_m\in T be distinct such that p(t_1,\dots,t_m)=0 for some p\in k[X_1,\dots,X_m]. We can view p(t_1,\dots,t_m) as a polynomial in t_m with coefficients in k[t_1,\dots,t_{m-1}], i.e. say

\displaystyle p(t_1,\dots,t_m)=\sum_{i=0}^n p_i(t_1,\dots,t_{m-1})t_m^i=0.

Since P(1) is true, it folows that p_i(t_1,\dots,t_{m-1})=0 for each i=0,1,\dots,n. But then p_i=0 for each i by our hypothesis. Thus p=0, implying P(m) is true.

Therefore, by induction, T is algebraically independent, i.e. T\in\mathcal F. \square


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Filed under Algebra, Set theory

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