Suppose that $p(x)=x^9+x^8+x^4+x^2+1 \in \mathbb{Z}_2[x]$. I have to show this polynomial is irreducible.
2026-04-04 11:58:16.1775303896
I have to show this polynomial is irreducible.
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If the polynomial is reducible, it must have an irreducible factor of degree at most $4$. It is clear by the Factor Theorem that $p$ has no factors of degree $1$. There is one irreducible quadratic in ${\Bbb Z}_2[x]$, there are two irreducible cubics and three irreducible quartics. Try them all.
Source for numbers: well known, or http://oeis.org/A001037.
Comment. As we are working in ${\Bbb Z}_2[x]$ with polynomials of degree less than $10$, you can do the calculations by counting on your fingers. This is not a joke, it really is possible - try it!!!