Decompose $x^4 + x^3 + 1$ into irreducible factors over $\mathbb{Z}_2$
I think that the given polynomial is already irreducible in $\mathbb{Z}_2$, therefore the only irreducible factors are $x^4 + x^3 + 1$ and $1$? Or am I missing something?
2026-04-06 07:06:41.1775459201
Decompose $x^4 + x^3 + 1$ into irreducible factors over $\mathbb{Z}_2$
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Yes, $x^4+x^3+1$ is irreducible. For it has no roots in $\mathbb{Z}_2$. And the only irreducible quadratic is $x^2+x+1$, which does not divide $x^4+x^3+1$.