I think I learned about this from a comment on MathOverflow.
Recall that the field of two elements is the ring
of integers modulo
. In other words, it consists of the elements
and
with addition
and the obvious multiplication. Clearly every nonzero element is invertible, so
is a field.
Lemma. The field is algebraically closed.
Proof. We need to show that every non-constant polynomial has a root. Suppose
does not have a root, so that
and
. Then
, so
is the constant polynomial
. This contradicts the assumption that
is non-constant.