TITLE: Let $F$ be a field of order $2^n$. Prove that characteristic of $F$ is 2.
QUESTION [7 upvotes]: I figure that Lagrange's theorem and the fact that the characteristic of an integral domain is either $0$ or prime should be used, but just can't figure it out exactly.

REPLY [5 votes]: By Cauchy's Theorem, the additive group of $\;\Bbb F\;$ has an element of order two, meaning
$$a+a=a(1+1)=0\implies 1+1=0$$
and clearly (why?) this is the minimal possible integer with this characteristic...