Degree of a splitting field over $ \mathbb{F}_{4}(t)$

Question

Let $ K=\mathbb{F}_{4}(t) $ and $ f(X)=X^{9}-t \in K[X] $, where $ t $ is transcendental over $ \mathbb{F}_{4} $. I am asked to determine the degree of the splitting field $ L $ of $ f $ over $ K $ and to show that that the Galois group $ G=Gal(L/K) $ contains a normal subgroup $ H $ isomorphic to $ \mathbb{Z}/9\mathbb{Z} $ with $ |G:H|=3 $, but $ G $ not isomorphic to $ \mathbb{Z}/3\mathbb{Z}\times \mathbb{Z}/9\mathbb{Z} $.

I know that $ [L:K] $ must divide 9! since $ f $ is irreducible over $ K $, but I don\t see how to determine its value. I also don't know how to approach the second question.

I would appreciate any help. Thank you!

Answer 1

Extended hints:

• Show that $f(X)$ is irreducible. Use Eisenstein in $\Bbb{F}_4[t]$ and the fact that $t$ is irreducible.
• Show that the ratio of any two zeros of $f(X)$ is a ninth root of unity.
• If $\zeta$ is a primitive ninth root of unity show that $\Bbb{F}_4[\zeta]=\Bbb{F}_{64}.$
• If $\alpha$ is one of the roots of $f(X)$ show that $L=K(\alpha,\zeta)$.
• Explain why $K(\zeta)/K$ and $L/K(\zeta)$ are Galois, but $K(\alpha)/K$ is not.
• Show that $\sigma:\alpha\mapsto\alpha\zeta$ generates $Gal(L/K(\zeta))$.
• Apply basic Galois correspondence to the previous two bullets to get what you need.

Answer 2

We have field extensions

$$K=\mathbb F_4(t) \subset M=\mathbb F_{64}(t) \subset L=\mathbb F_{64}(\sqrt[9]{t})$$ and the right hand side is the splitting field. In particular $\operatorname{Gal}(L/M)$ is a normal subgroup of $\operatorname{Gal}(L/K)$ of index $[M:K]=3$.

It is also well known that $\operatorname{Gal}(L/M) \cong \mathbb Z/9\mathbb Z$.

$\operatorname{Gal}(L/K)$ is not isomorphic to $\mathbb Z/9\mathbb Z \times \mathbb Z/3\mathbb Z$ because it is not abelian, because abelian extensions have the property, that adjoining one root automatically adjoins the other roots, too. This not the case for $L/K$.

Of course to get the non-abelian property, one can also note that the intermediate field $\mathbb F_4(\sqrt[9]{t})$ is not normal over $K$.