Algebraic Field Extension of Finite Field

Question

I have a question about finite field extensions over finite field $\mathbb{F}_p$ with characteristic $p > 0$:

It is well known from algebra that every finite field extension $\mathbb{F} \ | \ \mathbb{F}_p$ is exactly the field $\mathbb{F}_q$ where $q= p^n$ for $n=[\mathbb{F} \ | \ \mathbb{F}_p]$ or in other words $\mathbb{F}$ is the splitting field of the polynomial $X^q-X$.

Futhermore $\mathbb{F}$ has Galois Group isomorphic to cyclic group $\mathbb{Z}/n$ with generator $\sigma: a \to a^p$.

My questions:

1. is what happens with the field extension $\mathbb{F}_p(\zeta_n) \ | \ \mathbb{F}_p$ where $p \nmid n$ and $\zeta_n$ is the n-th primitive root?

Obviously here $\mathbb{F}_p(\zeta_n)$ is the splitting field of $X^n-1$ but because $n \neq p^r$ by assumpting this contradicts the considerations above, doesn't it?

2. What Galois Group $Gal(\mathbb{F}_p(\zeta_n) \ | \ \mathbb{F}_p)$ has $\mathbb{F}_p(\zeta_n)$ and why?

Answer 1

The polynomial $X^n-X$ is not irreducible over $\mathbb{F}_p$, you have to find the minimal polynomial of $\zeta_p$ to determine the degree of the extension.

Answer 2

The multiplicative group of a field $\mathbb{F}_q$ is cyclic of order $q-1$. So for $p\nmid n$, there is a primitive $n$th root of unity in $\mathbb{F}_q$ iff $n$ divides $q-1$. This means $\mathbb{F}_p(\zeta_n)$ is just $\mathbb{F}_q$ where $q$ is the least power of $p$ such that $n$ divides $q-1$. If $q=p^r$, its Galois group is then cyclic of order $r$. (This $r$ can also be described as the least $r>0$ such that $p^r$ is $1$ mod $n$; that is, the multiplicative order of $p$ mod $n$.)

Obviously here $\mathbb{F}_p(\zeta_n)$ is the splitting field of $X^n-X$ but because $n \neq p^r$ by assumpting this contradicts the considerations above, doesn't it?

This is no contradiction. A field extension can be the splitting field of many different polynomials. Every finite extension of $\mathbb{F}_p$ is the splitting field of a polynomial of the form $X^{p^r}-X$, but they can also be described as splitting fields of polynomials that are of different forms.