I've been struggling for hours and so far have shown the polynomial is irreducible, $p(0) \neq 0$, and monic.
All there is left to prove is that the polynomial is of order $15$.
Then I can use a theorem.
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Welcome to math SE. Have a look at mathjax for your mathematical expressions. – Alain Remillard Nov 17 '22 at 18:17
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1$\Phi_{15,F_2}(X)=X^8+X^7+X^5+X^4+X^3+X+1=(X^4+X+1)(X^4+X^3+1).$ – Anne Bauval Nov 17 '22 at 18:22
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3If $\alpha$ is a root of $p(x)$, then in $\mathbb{F}_2$, we can write $$\alpha^4=\alpha+1.$$ Now possible orders for $\alpha$ are $1,3,5,15$. See if you can rule out the lower orders. – Anurag A Nov 17 '22 at 18:28
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A brute force proof is a part of this old answer of mine. Proving primitivity doesn' t actually take that much because you only need to rule out five as a possible order. And the roots of unity of order five are the zeros of $x^4+x^3+x^2+x+1$. – Jyrki Lahtonen Nov 17 '22 at 20:57