$\mathbb{Q}$-linear combination of a polynomial in the field $\mathbb{Q}[x]/(x^3+2x+2)$

Question

I am doing my homework in field theory and I have the following task to accomplish. Given is the polynomial $f = x^3 + 2x + 2 \in \mathbb{Q}[x]$ and $\alpha$ is the image of $x$ in $K = \mathbb{Q}[x]/(f)$. I have already shown, that $f$ is irreducible, using Gauss' lemma and the fact that it suffices to show that $f$ is irreducible over $\mathbb{Z}[x]$. Then I used Eisenstein. I need to find a $\mathbb{Q}$-linear combination which expresses $\frac{2\alpha + 3}{\alpha + 1}$ in terms of $1, \alpha, \alpha^2$.

Answer 1

Apply the Euclidean algorithm to $x^3+2x+2$ and $x+1$.

Details:

As you have shown the former is irreducible, you will get $1$ expressed as a linear combination of $x^3+2x+2$ and $x+1$.

Then substitute $\alpha$ for $x$, and the $x^3+2x+2$ term disappears. You are left with $(\alpha+1)\beta=1$. So $\beta=\frac1{\alpha+1}$.

Finally multiply $\beta$ by $(2\alpha+3)$ and reduce modulo $f$ if necessary.