minimal polynomial over $\mathbb{Z}_7(t^{14})$

Question

I have ploblem with proving a certain polynomial is really minimal.

If $F$ is $\mathbb{Z}_7(t^{14})$, I should find minimal polynomial of $t^4$ and $t^5$ over $F$.

In my thought, I guess the answer is $x^7-t^{28}$ and $x^{14}-t^{70}$, respectively. But how can I prove that they are irreducible? Is there any idea?

Answer 1

I assume that you intend $t$ to be transcendental over $\mathbb{F}_7$.

$X^7-t^{28}=(X-t^4)^7$ in $\mathbb{F}_7(t)$.

So if $X^7-t^{28}$ factorises it has a factor $(X-t^4)^s$ for $0<s<7$.

But $(X-t^4)^s=X^s-s t^4 X^{s-1}+\dots $ and $st^4\not\in\mathbb{F}_7(t^{14})$.

The other one is similar.