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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 $\mathbb{F}_8={\overline{0},..,\overline{7}}$

John.W
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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 $\mathbf F_2[x]$?

KCd
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  • As far as I know $\mathbb{F}_2[X]$ has only 4 polynomials of which $x^2+x+\overline{1}$ is the only irreducible – John.W May 01 '23 at 23:34
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    @John.W $\Bbb F_2[X]$ is the ring of the polynomials with coefficients in $\Bbb F_2$, therefore it has infinitely many elements and, in fact, it has irreducible polynomials of every degree. – Sassatelli Giulio May 01 '23 at 23:40
  • @John.W the polynomials $x^n$ for $n \geq 1$ are infinitely many different polynomals, where this list includes one of each degree. – KCd May 01 '23 at 23:43
  • @KCd Oh yeah sorry, I was thinking about polynomials of second deg=2 (I don't know why, I was thinking about that) – John.W May 01 '23 at 23:45
  • @John.W You may be have in mind "polynomial functions”, there are $4$ of those, but there are infinitely many (formal) polynomials in $\Bbb F_2[X]$. – jjagmath May 01 '23 at 23:45
  • I did some reading as you suggestet, for p prime to get $\mathbb{F}{p^n}$ I need to calculate a irreducible polynomial $f(x)$ in $\mathbb{F}_p$, then $\mathbb{F}{p^n}=\mathbb{F}_p[x]/f(x)$ – John.W May 01 '23 at 23:58
  • But how do I know which irreducible polynomial to calculate? – John.W May 02 '23 at 00:00
  • Think about how big $\mathbf F_p[x]/(f(x))$ is in terms of $p$ and $\deg f$. Then set the size equal to $8$ to get information about $\deg f$. – KCd May 02 '23 at 00:05
  • @KCd So I know that $|\mathbb{F}_8|=8=2^3$ and thus $|F_2[x]/(f(x))|=8$. The coefficients of $F_2[X]$ can only be $\overline{0}$ or $\overline{1}$, this would mean that $f$ has to be of degree 3, right? – John.W May 02 '23 at 00:42
  • This would mean that I ether choose $x^3 +x +\overline{1}$ or $x^3+x^2+\overline{1}$ because those are the only irreducible polynomials in $\mathbb{F}_2[X]$? – John.W May 02 '23 at 00:59