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I'm trying a brute force approach, evaluating every single polynomial of degree 5. I was decomposing them using Ruffini in order to check if they are irriducible.

With this approach for example $x^5+x^4+1$ seems irreducible, but from the table it's not...

Is this approach valid? There is some better methods?

Alessar
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    Probable duplicate of https://math.stackexchange.com/questions/32197/find-all-irreducible-monic-polynomials-in-mathbbz-2x-with-degree-equal – lhf Dec 14 '18 at 11:27
  • Your polynomial is not irreducible over $\Bbb{F}_2$ because $$x^5+x^4+1=(x^5+x^4+x^3)+(x^3+1)=(x^2+x+1)x^3+(x^2+x+1)(x+1)=(x^2+x+1)(x^3+x+1).$$ – Jyrki Lahtonen Dec 14 '18 at 13:21
  • For degree five specifically, see this. For all low degree polynomials the suggestion of lhf is better. This is a dupe of one or the other, but I dare not cast my (immediately binding) vote. – Jyrki Lahtonen Dec 14 '18 at 13:24
  • Anyway, a degree five polynomial is irreducible if it has no linear or quadratic factors, so in the case of $\Bbb{F}_2$ it suffices to check that none of $x,x+1,x^2+x+1$ are factors. Equivalently, you can check that none $f(0),f(1),f(\omega)$ vanish. Here $\omega$ is a primitive root of unity of order three, a root of the only irreducible quadratic $x^2+x+1$. So, with $f(x)=x^5+x^4+1$ we can also calculate $$f(\omega)=\omega^5+\omega^4+1=\omega^2+\omega+1=0.$$ – Jyrki Lahtonen Dec 14 '18 at 13:27
  • See this nice post by Bill Dubuque for testing whether a given trinomial is a multiple of $x^2+x+1$. – Jyrki Lahtonen Dec 14 '18 at 13:29

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