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!
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. .