Compute a minimal polynomial over QQ

Suppose \theta is algebraic over \mathbb{Q} with minimal polynomial p(x) = x^3 - x^2 - x - 1. (p(x) is indeed irreducible since p(x-1) is Eisenstein at 2.) Compute the minimal polynomial of \eta = \theta^2+3 over \mathbb{Q}.


Note that \eta has degree at most 3 over \mathbb{Q}. Evidently, we have \eta^2 = 8\theta^2 + 2\theta + 10 and \eta^3 = 52\theta^2 + 24\theta + 40. Suppose \eta^3 + a\eta^2 + b\eta + c = 0; comparing coefficients, we have the following system of linear equations: 2a = -24, 8a+b = -52, and 10a+3b+c = -40. Evidently then \eta is a root of t(x) = x^3 - 12x^2 + 44x - 52. We can see using the rational root theorem that t(x) has no linear factors over \mathbb{Q}, and thus is irreducible. So t(x) is the minimal polynomial of \eta over \mathbb{Q}.

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