Compute the minimal polynomial of an algebraic number over QQ

Find the minimal polynomial of 1+i over \mathbb{Q}.


Note that 1+i \in \mathbb{Q}(i), and that \mathbb{Q}(i) has degree 2 over \mathbb{Q} (since i is a root of the irreducible x^2+1). That is, the degree of \mathbb{Q}(1+i) over \mathbb{Q} is at most 2.

Knowing an upper bound on the degree of the minimal polynomial of \alpha = 1+i, we can compute the powers of \alpha and solve the linear system \alpha^2 + a\alpha + b = 0.

Evidently, 1+i is a root of p(x) = x^2-2x+2. (WolframAlpha agrees.) Moreover, p(x) is irreducible over \mathbb{Q} by Eisenstein’s criterion; so it is the minimal polynomial of 1+i over \mathbb{Q}.

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