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.

Amazing! I’ve had trouble picturing the algebraic closure of finite fields in the past, but it’s great to have this simple example of a finite field that’s already algebraically closed!

Can you do F1 in a future post? I’ve wondered about that one too!