Finding primitive element of $\mathbb{F}_8/\mathbb{F}_2$

Question

I am trying to find a primitive element $\alpha$ of the field extension $\mathbb{F}_8 / \mathbb{F}_2$. Further I want to determine the minimal polynomial $m(\alpha,\mathbb{F}_2)$.

First I will write don't what I know:

$\mathbb{F}_p:=\mathbb{Z}/p\mathbb{Z}$ for $p$ prime

Let $L/K$ be a field extension,if there exists $\alpha \in L$ such $K(\alpha)=L$, then $\alpha$ is called a primitive element.

$\mathbb{F}_8=\mathbb{F}_2[X]/m(\alpha,\mathbb{F}_2)$

$\mathbb{F}_2=$ {$\overline{0},\overline{1}$}

Consider the Field extensio $F_8/F_2$

I know that if $\alpha$ is a primitive element, then $K(\alpha)= ${$ \frac{f(\alpha)}{g(\alpha)}: f,g \in K[X] $}

I have no clue how to even start. If someone could explain to me how to solve this I would be very thankful.

Edit: At the beginning I did the mistake to write $\mathbb{F}_8={\overline{0},..,\overline{7}}$

Answer 1

Watch out! The field $\mathbf F_8$ is not $\{\overline{0}, \overline{1},\ldots,\overline{7}\}$. For a prime power $p^n$ where $n > 1$, the integers mod $p^n$ are not a field.

I advise you to read your textbook or speak with your course instructor to learn how to build concrete examples of finite fields of non-prime size.

Do you know irreducible polynomials of low degree in $\mathbf F_2[x]$?