AAR3

From Exampleproblems

Determine whether $f(x) = x^4+15x+7\,$ is irreducible over $\mathbb{Q}\,$ or not.

Mapping this polynomial to a new field by way of $\sigma:\mathbb{Q}[x]\rightarrow \mathbb{Z}/(2)[x]\,$ gives $f(x)\mapsto f^\sigma(x) = x^4+x+1\,$ .

Check all elements in the new field for roots (linear factors): $f^\sigma(0) = f^\sigma(1) = 1\,$ so there are no linear factors.

$x^2\,$ is reducible. $x^2+1\,$ has 1 as a root. $x^2+x=x(x+1)\,$ The only irreducible quadratic in this field is $x^2+x+1\,$ . Polynomial long division will show that it does not divide into $x^4+x+1\,$ . Therefore $f^\sigma\,$ is irreducible over $\mathbb{Z}/(2)\,$ and $f\,$ is irreducible over $\mathbb{Q}\,$ .