## 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 $(1) = (\alpha,\beta)$.