Let $f(x)=x^4+x^3+4x-1\in\mathbb{Z}_5[x]$. Let $K$ be the splitting field of $f$ over $\mathbb{Z}_5[x]$. Find $\mathrm{Gal}(K/\mathbb{Z}_5)$.

Question

Let $f(x)=x^4+x^3+4x-1\in\mathbb{Z}_5[x]$. Let $K$ be the splitting field of $f$ over $\mathbb{Z}_5$. Find $\mathrm{Gal}(K/\mathbb{Z}_5)$.

So, the first thing we need to do is find the splitting field, and I am just sort of stuck. I know that $f(1)=0$, but I don't believe there are any more roots. So should I just use a primitive fourth root of unity? Any help is greatly appreciated! Thank you.

Answer 1

You have found a root, $f(1)=0$, but missed another one, $f(4)=0$. So, $f$ factors into the product of two linear polynomials (that we may ignore from hereon) and a quadratic polynomial $q$. Hence, the splitting field of $f$ is at most of degree $2$ (generated by either root of $q$) and hence, Galois with either Galois group $\mathbb Z/2\mathbb Z$ or trivial group (if $q$ splits over $\mathbb Z_5$ as well).

We have the following:

$$ f(x)=x^4+x^3+4x-1\equiv(x-1)(x-4)(x^2+x+1)\mod5 $$

So $q(x)=x^2+x+1$, which is irreducible over $\mathbb Z_5$ (as it has no roots). So $[K:\mathbb Z_5]=2$ and the rest follows as sketched above.