I have this polynomial:
$x^8+x^4+x^3+x+1$
and I would like to know if it is irreducible over $\mathbb{F}_q$ with $q=2^8$. My book gives me it is irreducible but matlab says it is not irreducible.
I have this polynomial:
$x^8+x^4+x^3+x+1$
and I would like to know if it is irreducible over $\mathbb{F}_q$ with $q=2^8$. My book gives me it is irreducible but matlab says it is not irreducible.
Theorem. Let $P$ be any polynomial of degree $k\ge 2$ with coefficients in $\mathbb F_p$. Then $P$ is reducible over $\mathrm{GF}(p^k)$.
Proof. If $P$ is reducible over $\mathbb F_p$, then the same factorization works over $\mathrm{GF}(p^k)$. So assume $P$ is irreducible over $\mathbb F_p$. Then $\mathbb F_p[X]/(P)$ is isomorphic to $\mathrm{GF}(p^k)$, so the image of $X$ under this isomorphism is a root of $P$. Therefore $P$ has a linear factor in $\mathrm{GF}(p^k)[X]$.
If a polynomial of degree 8 over $\mathbb{F}_q$ is irreducible over the finite field $\mathbb{F}_q$, then its splitting field is $\mathbb{F}_{q^8}$, and is thus reducible over $\mathbb{F}_{q^8}$. If a polynomial of degree 8 over $\mathbb{F}_q$ is reducible over the finite field $\mathbb{F}_q$, then it remains reducible over $\mathbb{F}_{q^8}$.