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.

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