Prove the extension $\frac{\mathbb Z_7[x]}{\langle 1+x+x^3\rangle}:\mathbb Z_7$ is normal

Question

Is this solution correct? It's actually just the structure of the solution.

Prove that the extension $\frac{\mathbb Z_7[x]}{\langle 1+x+x^3\rangle}:\mathbb Z_7$ is normal.

Proof:

Clearly the extension is Galois (I proved with a lot of details in my exam and within the proof of that, I mention that $\left[\dfrac{\mathbb Z_7[x]}{\langle 1+x+x^3\rangle}:\mathbb Z_7\right]=3).$ So I have a finite Galois extension. Now, by Theorem*, $\dfrac{\mathbb Z_7[x]}{\langle 1+x+x^3\rangle}:\mathbb Z_7$ is normal. ∎

Theorem*: All $K/F$ finite, normal and separable is Galois.

Notes:

My professor told me that this solved exercise made by me is wrong, because 'I applied the Theorem* in a wrong way'. However the Theorem* becomes an if and only if when the extension $K/F$ is finite (which is my case, 3).

Definition: An extension $K/F$ is Galois if $G(K/F)^+=\sigma(F),$ where $\sigma:F\to K$ is a monomorphism. (the fixed field characterization)

Definition: An extension $K/F$ is normal if every irreducible polynomial in $F[x]$ with a root on $K$, splits on $K$.

The Theorem* was proved in class, however this side $\gets$ of the Theorem* was not proven, but in advisories my professor told me about it: Galois=normal+separable if $K/F$ is finite.

Did I miss something this time? :)

Answer 1

Your statement of Theorem* is imprecise and grammatically incorrect. From context, it appears to me that the precise statement is:

Let $K/F$ be a field extension. Then if $K/F$ is finite, normal, and separable, then it is Galois.

Assuming this is the statement of Theorem*, then is incorrect to say "by Theorem*" in your proof, because you are not using Theorem*. Instead, you are using the converse of Theorem* (assuming the extension is finite), which is a different theorem. It is fine to use the converse, but you need to accurately say which theorem you are using to have a correct proof.

Alternatively, perhaps the statement of Theorem* is:

Let $K/F$ be a finite field extension. Then $K/F$ is normal and separable if and only if it is Galois.

In that case, your proof is correct, since both directions are included in the statement of Theorem*.