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Describing Galois groups of some local fields (Ignore this quoted question, it is not related/ important)

Following my question, I have some questions in my mind. Consider this special case:

Let $p$ be a prime number, and consider finite fields $\mathbb{F}_p$ and $\mathbb{F}_{p^2}$, i.e fields of cardinality $p$ and $p^2$. I know that $\mathbb{F}_{p^2}\cong\mathbb{F}_{p}[x]/(x^{p^2}-x)\cong\mathbb{F}_{p}(\zeta_{p^2-1})$, and $[\mathbb{F}_{p^2}:\mathbb{F}_{p}]=2$, and the only nontrival element of the $Gal (\mathbb{F}_{p^2}/\mathbb{F}_{p})$ is $[\zeta_{p^2-1}\mapsto \zeta_{p^2-1}^p]$, and we will denote the two elements of the Galois group by $id$ and $\sigma$.

Every element of $\mathbb{F}_{p^2}$ has a minimal polynomial of degree $1$ or $2$ over $\mathbb{F}_{p}$ (because $[\mathbb{F}_{p^2}:\mathbb{F}_{p}]=2$).

What is the minimal polynomial of random elements of $\mathbb{F}_{p^2}$? (Ignore the elements with a minimal polynomial of degree one)

Especially what is the minimal polynomial of $\zeta_{p^2-1}$? Is the following attempt true?

My attempt: $id(\zeta_{p^2-1})=\zeta_{p^2-1}$ and $\sigma(\zeta_{p^2-1})=\zeta_{p^2-1}^p$. Then the minimal polynomial of $\zeta_{p^2-1}$ is $$\left(x-id(\zeta_{p^2-1})\right)\left(x-\sigma(\zeta_{p^2-1})\right)=\left(x-\zeta_{p^2-1}\right)\left(x-\zeta_{p^2-1}^p\right) \\=x^2-(\zeta_{p^2-1}+\zeta_{p^2-1}^p)x+\zeta_{p^2-1}\zeta_{p^2-1}^p=x^2-(\zeta_{p^2-1}+\zeta_{p^2-1}^p)+\zeta_{p-1},$$ We can see the coefficent $\zeta_{p-1}$ as a primitive root modulue $p$; i.e. a generating element of $\mathbb{F}_p^\star$, and we can consider it as an element of $\mathbb{F}_p$.

But what can we say about the coefficent $\zeta_{p^2-1}+\zeta_{p^2-1}^p$? How can we conclude that $\zeta_{p^2-1}+\zeta_{p^2-1}^p \in \mathbb{F}_p$?

Let $p=5$. Can we compute the minimal polynomial of $\zeta_{5^2-1}$? What are the coefficents $\zeta_{5^2-1}+\zeta_{5^2-1}^5$ and $\zeta_{5^2-1}\zeta_{5^2-1}^5=\zeta_{5-1}$? I can say that $\zeta_{5^2-1}\zeta_{5^2-1}^5=\zeta_{5-1}=i$ is either $2$ or $3$.

RobPratt
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Tireless and hardworking
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    The quotient ring $\Bbb{F}_p[x]/\langle x^{p^2}-x\rangle$ is not a field. It is a ring that has dimension $p^2$ over the prime field. The reason is that $x^{p^2}-x$ is not irreducible but rather the product of all the irreducible monic and quadratic polynomials. – Jyrki Lahtonen Jun 12 '22 at 19:36
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    The next to last question is answered by Galois theory. $\zeta+\zeta^p$ is fixed by the Frobenius automorphism, and hence is an element of the prime field. – Jyrki Lahtonen Jun 12 '22 at 19:38
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    The answer to the last question depends. In a cyclic group of order $24$ there are $\phi(24)=8$ elements that generate the group. Therefore there are $8$ primitive roots of unity of order $24$. Between them, they have four quadratic minimal polynomials, each conjugate pair of primitive elements sharing the same minimal polynomial. Two of them correspond to $\zeta^{p+1}=2$ and the other two to $\zeta^{p+1}$. An extra observation we can make is that if the constant term of the minimal polynomial of $\zeta$ is $2$, then the minimal polynomial of $\zeta^{-1}$ has constant term $3$. – Jyrki Lahtonen Jun 12 '22 at 19:42
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    I'm fairly sure that individually these questions have been answered already on the site. I will try to find something, but do search yourself, too :-) – Jyrki Lahtonen Jun 12 '22 at 19:44
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    Last but not least. Yes, you have a correct recipe for the minimal polynomial. Its roots are each others images under Frobenius. Good job there. – Jyrki Lahtonen Jun 12 '22 at 19:45
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    The first hits: 1,2,3. The accepted answer to the second link lists the minimal polynomials of the primitive 24th roots of unity over $\Bbb{F}_5$. – Jyrki Lahtonen Jun 12 '22 at 19:49
  • @JyrkiLahtonen thanks for your patience and all of your helpful comments, your comments solved everything for me. – Tireless and hardworking Jun 12 '22 at 20:02
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    Glad to hear that. If there is something not covered in the linked threads, or you want some detail of an argument checked out, you are welcome to post that as an answer, and hopefully get more feedback. I visit all the questions tagged [tag:finite-fields], but cannot promise a schedule :-) If all is clear, we can close this as a duplicate of the others. But there is no particular rush for such action. – Jyrki Lahtonen Jun 12 '22 at 20:05
  • Dear @JyrkiLahtonen thanks for your supportive comments and for your generous kindness. After your comments everything was clear to me, and to me, your comments are enough and answer everything, and to close this question maybe we can regard it as a duplicate of the others, perhaps this question: https://math.stackexchange.com/questions/2744985/show-2-alpha-is-a-primitive-root-of-mathbbf-25 or another question – Tireless and hardworking Jun 13 '22 at 05:40

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