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List all polynomials in $\mathbb F_2[x]$ that are minimal polynomials of elements from $\mathbb F_{16}$.

Since minimal polynomials are irreducible, this problem just asks to list irreducible polynomials of certain degrees over $\mathbb F_2$. But I have a hard time realizing what degrees I should consider. Does it have to do with the fact about the orders of subfields of finite fields? I'm not sure how to apply it.

user557
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    All elements of $\mathbb{F}{16}$ are roots of $x^{16}-x$. Therefore, all minimal polynomials of elements of $\mathbb{F}{16}$ are irreducible factors of $x^{16}-x$. –  Aug 06 '18 at 01:48
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    Don’t forget that the degree of the minimal polynomial for $\alpha$ is the degree of $\Bbb F_2(\alpha)$ over $\Bbb F_2$. So this degree can only be $1$, $2$, or $4$. Thus aside from $x$ and $x+1$, the only irreducible linears, and $x^2+x+1$, the only irreducible quadratic, you need to find the three irreducible quartics. This is not hard, when you realize that the constant term must be $1$ and there must be an odd number of monomials. – Lubin Aug 06 '18 at 02:02
  • After Lubin's comment the question is reduced to this. Opinions differ whether that is now a duplicate or not. I would say it is, but I don't want to be the first to cast a vote, because I have dupehammer powers here. – Jyrki Lahtonen Aug 06 '18 at 07:31
  • Of course, spiralstotheleft's comment reduces the question to this instead. May be you could post an answer based on those? That way you get a bit more feedback on your understanding this theme, and we get this question out of the unanswered pile. – Jyrki Lahtonen Aug 06 '18 at 07:32
  • @JyrkiLahtonen I've posted an answer below. I didn't look at the link your first comment until after I've posted the answer; now I see that my answer is in fact a duplicate of the answer by Jorge Fernández in that topic. – user557 Aug 06 '18 at 21:29

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The elements of $\mathbb F_{2^4}$ are the roots of $x^{2^4}-x$. By the general theory, the irreducible factors of $x^{p^r}-x$ over $\mathbb F_p$ are the irreducible polynomials in $\mathbb F_p[x]$ whose degrees divide $r$. Thus we need to list all irreducible polynomials over $\mathbb F_2$ of degrees $1,2,4$.

Degree $1$: $x,x+1$.

Degree $2$: $x^2+x+1$ (the other three polynomials have roots in the field).

Degree $4$: write out all polynomials of degree $4$ (there are $16 $ of them) and cross out those having roots. The remaining polynomials are $$x^4+x^3+x^2+x+1,x^4+x^2+1,x^4+x+1,x^4+x^3+1.$$ We need to delete from this list the polynomials that have an irreducible factor of degree $2$. Since the only irreducible polynomial of degree $2$ is $x^2+x+1$ and $(x^2+x+1)^2=x^4+x^2+1$, the three polynomials $x^4+x^3+x^2+x+1,x^4+x+1,x^4+x^3+1$ form a complete list of degree $4$ irreducible polynomials.

user557
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