Show that a given algebraic integer is a unit in the ring of integers in the field extension it generates

Let \alpha be a root of p(x) = x^{17} - 43x^6 + 1. Show that, as ideals in the ring \mathcal{O} of integers in \mathbb{Q}(\alpha), (\alpha) = \mathcal{O}.


Note that p(x) = h\prod q_i(x) is a product of the minimal polynomials of its roots. Since p(x) is monic, in fact h = 1. Since the constant coefficient of p(x) is 1, the constant coefficient of the minimal polynomial q(x) for \alpha over \mathbb{Q} is 1. That is, N(\alpha) = 1. By Theorem 7.3 in TAN, \alpha is a unit; hence (\alpha) = \mathcal{O}.

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