TITLE: Finite field, every element is a square implies char equal 2
QUESTION [12 upvotes]: If $F$ is a finite field such that every element is a square, why must $char(F)=2$?

REPLY [3 votes]: Another way: all elts of a finite field of size $2n\!+\!1$ are squares $\Rightarrow \color{#0a0}{x^{n}\! - 1} $ has $2n$ roots (all elts $\ne\! 0$), contra a polynomial over a field has no more roots than its degree, by $\,a = b^2\Rightarrow  \color{#0a0}{a^n} = b^{\color{#c00}{2n}} \color{#0a0}{= 1},\,$ by Lagrange, since its multiplicative group has size $\,\color{#c00}{2n}$ (i.e. Euler's Criterion when size is prime).