I am being asked to find the following:
Let $F$ denote the field $\dfrac{\mathbb{F}_2[\alpha]}{(\alpha^3 + \alpha + 1)}$. Simplify $\alpha(\alpha + 1)(\alpha + 1)$ in $F$ and calculate $\alpha^{-1}$ in $F$.
I am trying to understand what exactly $\dfrac{\mathbb{F}_2[\alpha]}{(\alpha^3 + \alpha + 1)}$ means. Does it mean the set of all polynomials with variable $\alpha$ of the form $(\alpha^3 + \alpha + 1)$? If so, would it be right that I simply have $\alpha(\alpha + 1)(\alpha + 1) = \alpha^3 + 2\alpha^2 + \alpha = \alpha^3 + \alpha$ since $2\alpha^2 = 0$ since we are working in $\mod 2$? I am also unsure as to how to even begin calculating $\alpha^{-1}$.
Thank you in advance for your help.