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I'm trying to solve this question: Let $F=\mathbb{F}_2[y]/(y^3+y+1)$.

  1. Show that $F$ is a field
  2. Factorize $x^3+x+1$ and $x^3+x^2+1$ into irreducible monic polynomials in $F[X]$.

I managed to prove the first part using $\mathbb{F}_2$ is a field $\implies \mathbb{F}_2[y]$ is a PID $\implies$ an ideal is maximal iff it is generated by an irreducible element and then showing that $y^3+y+1$ is irreducible. However, I'm stuck on the second part. Any help is appreciated. Is there a general method for tackling this sort of questions?

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1 Answer 1

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We have $F=\mathbb F_2[\alpha]$, where $\alpha^3+\alpha+1=0$. Moreover, $F$ is a field with $8$ elements, all of the form $u\alpha^2+v\alpha+w$, with $u,v,w\in\mathbb F_2$. To factor those polynomials, just find which elements are roots of each.

For instance, $0=(\alpha^3+\alpha+1)^2=(\alpha^2)^3+\alpha^2+1$ and so $\alpha^2$ is another root of $x^3+x+1$. Divide $x^3+x+1$ by $x-\alpha$ and by $x-\alpha^2$ to find the third root.

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    $\begingroup$ To simplify the computations, one may observe that $x-\alpha=x+\alpha$, and similarly for $x-\alpha^2$. $\endgroup$
    – Bernard
    Nov 27, 2020 at 13:34
  • $\begingroup$ Thank you so much for your answer. However, I'm new to polynomial rings, so I don't completely get it. How do you know that $F=\mathbb{F}_2[\alpha]$? $\endgroup$
    – 14159
    Nov 28, 2020 at 17:26

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