Compute the minimal polynomial of a given element over the rationals

Compute the minimal polynomial of 1 + \sqrt[3]{2} over \mathbb{Q}.


Let \eta = 1 + \sqrt[3]{2}. Evidently, \eta^2 = 1 + 2 \sqrt[3]{2} + \sqrt[3]{2^2} and \eta^3 = 3 + 3 \sqrt[3]{2} + 3 \sqrt[3]{2^2}. Evidently, \eta^3 - 3\eta^2 = -3 \sqrt[3]{2}, so that \eta^3 - 3\eta^2 - 3 = -3\eta. Hence \eta is a root of p(x) = x^3 - 3x^2 + 3x - 3. This polynomial is Eisenstein at 3, hence irreducible over \mathbb{Q}. By this previous exercise, p(x) is the minimal polynomial of \eta over \mathbb{Q}.

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