Irreducible polynomial modulo 2

Question

I need to prove that polynomial $f(x) = x^{10}+x^{3}+1$ is irreducible modulo $2$. It is irreducible if $f|x^{1024}-x$, isn't it? I can use polynomial long division to check it, but this is not elegant. Is there any other way? Thanks.

Answer 1

You do not need to go to the polynomial $x^{2^{10}}-x$ which, by the way, equals $$x^{2^{10}}-x = \prod_{f \textrm{ irred} \deg | 10} f $$ (so that would work only in the opposite direction in fact), rather, show that it does not have any factor of degree $5$, $4$, $3$, $2$, $1$, that is, show that the gcd of this polynomial with each of the polynomials $$ x^{2^5}-x,\ x^{2^4}-x,\ x^{2^3} - x$$ is $1$, which is not too hard.