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.
.