is the field extension $\mathbb{Z}_{3}(X)\supseteq \mathbb{Z}_{3}(X^2)$ normal

Question

Is this field extension:

$\mathbb{Z}_{3}(X)\supseteq \mathbb{Z}_{3}(X^2)$

normal?

I pick $t^2-X^2\in \mathbb{Z}_{3}(X^2)$ which is irreducible over $\mathbb{Z}_{3}(X^2)$ as it's zeros are $t=\pm X$ and $\pm X\not\in \mathbb{Z}_{3}(X^2)$ and both roots belong to $\mathbb{Z}_{3}(X)$ so the extension is normal.

Is this okay?

Answer 1

That's what I would write down on my assignment.