I have 2-problems for help:
Given a prime $p$ and a positive integer $n$:
Problem (1): Show that there is an Abelian extension [i.e., Galois with Abelian Galois group] $K$ of $\mathbb{Q}$ with $[K:\mathbb{Q}] =p^{n}$.
Solution for Problem (1) (my attempt): We want to find a minimal (irreducible) polynomial $f \in \mathbb{Q}[x]$ such that $deg(f) = p^{n}$ and $Aut_{\mathbb{Q}} \big( \mathbb{Q}(\omega) \big)$ is abelian where $\omega$ is a root of $f$. We know the following theorems:
- Theorem(A): If the field $F$ contains once primitive $n-$root of unity and $E$ is the splitting field of the $f(x) = x^{n} - a \in F[x]$, then $Aut_{F}(E)$ embedded to $(\mathbb{Z}_{n} , +)$. Moreover, it is true the followings: $Aut_{F}(E) \cong \mathbb{Z}_{n} \Longrightarrow f(x)$ is irreducible $\Longrightarrow [ E : F ] = n$.
- Theorem(B): Let $F \subseteq \mathbb{C}$ be a field containing all the $n-$roots of unity over $\mathbb{Q}$. An extension $E/F$ is cyclic of order $n$ if and only if $E = F \big( \sqrt[n]{a} \big), for some $a \in F$.
Let $\omega$ be the primitive $p^n-$th root of unity. Then $K=\mathbb{Q}(\omega)$ is Galois extension of $\mathbb{Q}$ and abelian over $\mathbb{Q}$, as a cyclotomic extension of order $p^{n}$.
Am I correct? Please, tell me the correct solution without gaps and the idea behind the problem, thank you!
Problem (2): Let $F$ be a field with $char(F)=p$ and $|F| = p^r$ elements. If $K$ is a finite extension of $F$ with $K=F[\alpha]$, for some $\alpha \in K$, and $f$ is the minimal polynomial of $\alpha$ over $F$. Show that if $\beta$ is another root of $f$, then $\beta \in K$ and $\beta=\alpha^{p^{k}}$ for some $k \in \mathbb{Z}$.
Solution for Problem (2) (my attempt): We have that $f \in F[x]$ is the minimal polynomial of $\alpha$, so $[K : F] = deg(f)$ and $f(\beta)=0$.
Since $F$ is a finite field with $|F|= p^{r}$, then: $y \in F \Longleftrightarrow y^{p^{r}} = y$.
Also, $\alpha$ is a root of the polynomial $x^{p^{r}} - \alpha^{p^{r}} = ( x - \alpha )^{p^{r}} \in K[x]$, so $deg(f)$ divides $p^{r} \Longrightarrow deg(f) = p^{n}$ for some $n \leqslant r$, and then $[K:F] = deg(f) = p^{n}$.
How can we prove that: $\beta \in K$ and $\beta=\alpha^{p^{k}}$ for some $k \in \mathbb{Z}$?
Can you tell me the solutions of the problems? and also to tell me the ideas behind the problems!