April | 2021 | Lovely little lemmas

I think I learned about this from a comment on MathOverflow.

Recall that the field of two elements $\mathbf F_2$ is the ring $\mathbf Z/2\mathbf Z$ of integers modulo $2$ . In other words, it consists of the elements $0$ and $1$ with addition $1+1 = 0$ and the obvious multiplication. Clearly every nonzero element is invertible, so $\mathbf F_2$ is a field.

Lemma. The field $\mathbf F_2$ is algebraically closed.

Proof. We need to show that every non-constant polynomial $f \in \mathbf F_2[x]$ has a root. Suppose $f$ does not have a root, so that $f(0) \neq 0$ and $f(1) \neq 0$ . Then $f(0) = f(1) = 1$ , so $f$ is the constant polynomial $1$ . This contradicts the assumption that $f$ is non-constant. $\qedsymbol$

Lovely little lemmas

Or maybe ‘amusing observations’

Monthly Archives: April 2021

Algebraic closure of the field of two elements