When working in the field $\mathbb{Q}[X] / \langle X^3 + 3X + 3 \rangle$, let $a$ represent the image of $X$ under the natural quotient mapping.
I am trying to understand the range of strategies that I might use to represent the element $\frac{1}{a^2 +1}$ in the form $k_2 a^2 + k_1 a + k_0$ with $k_i \in \mathbb{Q}$. I have already determined the following:
- $\frac{1}{a}$ is $-\frac{1}{3} a^2 - 1$, an answer which I found by rearranging the equation $a^3 + 3a + 3 = 0$.
- $\frac{1}{a+1}$ is $a^2 - a + 4$, which was found by dividing $a^3 + 3a + 4$ by $a+1$.
In order to find $\frac{1}{a^2 + 1}$, I am attempting to use the same method. I started by rearranging $a^3 + 3a + 3 = 0$ to form the equation $\frac{1}{a^2 + 1} = \frac {a}{-2a-3}$ and proceeding with more division.
I have checked my work with Sage. The correct representation is $\frac{2}{13}a^2 - \frac{3}{13}a + \frac{4}{13}$, but the correct path forward is eluding me at the moment. Any possible strategy hints would be welcome. Thank you so much for your time!