How many distinct roots within an algebraic closure

Question

Let $E=\overline{F_2}$. How to find the number of distinct roots of $f(x)=x^{81}-1\in F_2[x]$ in $E$?

So far as I tried, I factorised $f$ into $$f(x)=(x-1)(x^{80}+x^{79}+\cdots+x+1)=(x-1)g(x)$$ And the problem became finding the number of distinct roots of $g(x)$ in $E$.

Of course this is a silly approach because I only looked the root in $F_2$ while all the other roots don't lie in it! But when I instead look at a bigger field, it seems complicated to express their elements via minimal polynomial over $F_2$ (due to difficulty in judging irreducibility).

Is this problem a trivial one (by which I mean it can be solved by hand) or does it have to be approached with advanced techniques? I'm only learning basic level algebra.

Answer 1

if $u$ is a multiple root, $f(u)=f'(u)=0$, $f'(x)=81x^{80}$ has only zero as root, but zero is not a root of $f$ so $f$ does not have multiple roots.