Let $\mathbb{F}$ be a field with $8$ elements und $k$ a prime field. ($1$st task) Determine the degree of the field extension $\mathbb{F}/k$. ($2$nd task) Is there an $a \in \mathbb{F} \backslash k $ with $a^3\in k$ so that the following is true $\mathbb{F}=k(a)$?
As I am currently studying for my algebra exam I decided to use old algebra exam exercises to practice, but this exercise is rather tough.
- First task
So my answer for this is that for a finite field, there are $p^r$ elements with $r\in \mathbb{N}$, where $p$ is a prime number. So logically speakin we have:
$p^r=8 \rightarrow p=2; r=3$
And the the degree of the field extension should be defined as
$\mathbb{F}/k:=\text{dim}_\mathbb{K}(\mathbb{F})=\{3\}$
Is this correct?
- Second task
Unfortunately I don't know how to begin solving this, so maybe someone could give me hint? Thanks in advance!