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Let $p(x)=x^4+x^3+1$ . Is $p$ irreducible in:

1) $\mathbb{F}_2$ ?

2) $\mathbb{F}_4$ ?

3) $\mathbb{F}_8$ ?

4) $\mathbb{F}_{16}$ ?

5) $\mathbb{F}_{32}$ ?

6) $\mathbb{F}_{64}$ ?

I can prove that $p$ is irreducible in $\mathbb{F}_2$ and $\mathbb{F}_4$ by showing it has no roots and no quadratic factors.

How can I complete questions 3-8?

convexboi
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    Try thinking about the structures of these fields and how roots of polynomials generate them from each other. For example, you know that $p(x)$ is an irreducible polynomial of degree $4$ over $\Bbb F_2$. How does that relate to a construction of the field $\Bbb F_{16}$? – Greg Martin Mar 16 '20 at 19:06
  • I am not really sure, like $\mathbb{F}_{16}$ can be obtained as an extension $\mathbb{F}_2(a)$ with $p(a)=0$? Could you elaborate a bit more? – convexboi Mar 16 '20 at 19:21

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