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.