## Compute an inverse in a given extension of QQ

Let $E = \mathbb{Q}(\sqrt[7]{3})$. Compute the multiplicative inverse of $1 + \sqrt[7]{3}$ in $E$.

We know that every element of $E$ has the form $\tau = \sum_{i=0}^6 \alpha_i \sqrt[7]{3}^i$. Setting $\tau(1 + \sqrt[7]{3}) = 1$ and comparing coefficients, we have the following system of equations: $\alpha_0 + 3\alpha_7 = 1$, $\alpha_0 + \alpha_1 = 0$, $\alpha_1 + \alpha_2 = 0$, $\alpha_2 + \alpha_3 = 0$, $\alpha_3 + \alpha_4 = 0$, $\alpha_4 + \alpha_5 = 0$, $\alpha_5 + \alpha_6 = 0$, and $\alpha_6 + \alpha_7 = 0$. Evidently this system has the solution $\alpha_0 = \alpha_2 = \alpha_4 = \alpha_6 = 1/4$ and $\alpha_1 = \alpha_3 = \alpha_5 = -1/4$.

Indeed, we can verify that $(1 + \sqrt[7]{3})(\frac{1}{4} \sum_{1=0}^6 (-1)^i \sqrt[7]{3}^i) = 1$.