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 has an upper bound in , then contains a maximal element.

Let be a field extension. We call a subset *algebraically independent* if for any and , implies , where is a polynomial. A maximal (with respect to ) algebraically independent subset is called a *transcendence base*.

**Theorem.** Every field extension has a transcendence base.

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

.

It suffices to show that , since then would be an upper bound for in .

Let be the statement:

If are distinct, and for some , then .

If , then for some . Since is algebraically independent, implies in . Thus is true.

Suppose is true. Let be distinct such that for some . We can view as a polynomial in with coefficients in , i.e. say

.

Since is true, it folows that for each . But then for each by our hypothesis. Thus , implying is true.

Therefore, by induction, is algebraically independent, i.e. .