In an algebraic integer ring, two elements whose norms are relatively prime generate the whole ring

Let \mathcal{O} be the ring of algebraic integers in a number field K. Suppose \alpha,\beta \in \mathcal{O} such that N(\alpha) and N(\beta) are relatively prime. Prove that (\alpha,\beta) = (1).


Recall that N(\alpha) and N(\beta) are rational integers. By Bezout’s identity, there exist rational integers h and k such that hN(\alpha) + kN(\beta) = 1. Note that N(\alpha)/\alpha is an algebraic integer (being a product of algebraic integers) and is in K; hence N(\alpha)/\alpha \in \mathcal{O}. Likewise, N(\beta)/\beta \in \mathcal{O}. So h(N(\alpha)/\alpha) \alpha + k(N(\beta)/\beta) \beta = 1, and we have (1) \subseteq (\alpha,\beta). Hence (1) = (\alpha,\beta).

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