Compute the degree of a given extension of the rationals

Compute the degrees of \mathbb{Q}(\sqrt{3+4i} + \sqrt{3-4i}) and \mathbb{Q}(\sqrt{1+\sqrt{-3}} + \sqrt{1-\sqrt{-3}}) over \mathbb{Q}.


Let \zeta = \sqrt{3+4i} + \sqrt{3-4i}. Evidently, \zeta^2 = 16. (WolframAlpha agrees.) That is, \zeta is a root of p(x) = x^2 - 16 = (x+4)(x-4). Thus the minimal polynomial of \zeta over \mathbb{Q} has degree 1, and the extension \mathbb{Q}(\zeta) has degree 1 over \mathbb{Q}.

Similarly, let \eta = \sqrt{1+\sqrt{-3}} + \sqrt{1-\sqrt{-3}}. Evidently, \eta^2 = 6 (WolframAlpha agrees), so that \eta is a root of q(x) = x^2-6. q is irreducible by Eisenstein’s criterion, and so is the minimal polynomial of \eta over \mathbb{Q}. The degree of \mathbb{Q}(\eta) over \mathbb{Q} is thus 2.

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