Show that $x^4+x+1$ is a prime polynomial over $F_2$

Question

To prove this, the only idea I had was to divide $x^4+x+1$ by all polynomials over $F_2$ and show that it cannot be evenly divided.

This seems tedious and I am sure there is a more elegant way to prove this. Can anyone give hints or guide me towards a proof?

Answer 1

Hint If a quartic polynomial factors over some field, then it has a linear factor and/or an irreducible quadratic factor. Over $F_2$ there are only two (strictly) linear polynomials and only one irreducible quadratic polynomial, namely $x^2 + x + 1$, and these facts together reduce the checking to just a few cases.

Answer 2

There is no root, and we need a constant term $1.$ We may simply calculate $$ \left(x^2 + 1 \right) \left(x^2 + 1 \right) $$ $$ \left(x^2 +x+ 1 \right) \left(x^2 +x+ 1 \right) $$ $$ \left(x^2 + 1 \right) \left(x^2 +x+ 1 \right) $$