How do I know the number of primitive elements, which are them and the degree of extensions of Galois Groups

Question

I am studying Galois Group and, I found some difficulties with exercises. I would like some explanation about how to understand:

1) How do I find the primitives elements of a Galois Group? Which procedures of calculation have I to do?

2) Which is the number of primitives elements in a Galois Group?

3) How do I make calculations like: $[GF(16):GF(4)]$, and more generally, $[GF(p^n):GF(p^r)]$ where $r$ divides $n$.

Thanks a lot!

Answer 1

$\newcommand{\GF}{\textrm{GF}}$$\renewcommand{\phi}{\varphi}$You mean Galois fields, right?

As correctly noted by Jyrki Lahtonen in his comments (thanks a bunch!) there are two notions of primitive element here. I had first written an answer for the meaning an element whose powers account for the whole multiplicative group.

In this case the answer to point 2 is

2. The number of primitive elements of the field $\GF(p^{n})$ is $\phi(p^{n} -1)$.

Then I thought that maybe it is the other definition we are talking about here, that is, an element $\alpha$ such that $\GF(p)[\alpha] = \GF(p^{n})$, and thus replaced the previous answer by

2. The number of primitive elements of the field $\GF(p^{n})$ is the degree of the polynomial $\prod_{d \mid n} (x^{p^{d}} - x)^{\mu(n/d)}$. Here $\mu$ is the Moebius function. Thus the number is $\sum_{d \mid n} p^{d} \mu(n/d)$.

3. If $r \mid n$, then $\lvert \GF(p^{n}) : \GF(p^{r}) \rvert = n/r$.