# Factorizing Polynomial in a Quotient Ring

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?

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.
• To simplify the computations, one may observe that $x-\alpha=x+\alpha$, and similarly for $x-\alpha^2$. Nov 27, 2020 at 13:34
• 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 $F=\mathbb{F}_2[\alpha]$? Nov 28, 2020 at 17:26