I am reading a book about algebraic number theory and I'm wondering whether the following statement is true:
Let $K$ be a number field and $\mathcal{O}_K$ its algebraic integers. Let $\beta \in \mathcal{O}_K$, is it true that $N(\beta)/\beta \in \mathcal{O}_K$? ($N$ is the norm of K over $\mathbb{Q}$)
If $K$ is a Galois extension of $\mathbb{Q}$ then we have that the norm is just the product of all Galois conjugates of a given element. Thus $N(\beta)/\beta$ would be the product of all Galois conjugates of $\beta$ except the one corresponding to the identity. Since Galois conjugates of an algebraic integer are also algebraic integers, the above follows.
Now I'm struggling with what happens when $K$ is only a separable extension. (separable always holds since we are in characteristic 0)
Best Answer
This still works. Let $L$ be the Galois closure of $K$ over $\Bbb Q$. Then $N(\beta)/\beta$ is a product of conjugates of $\beta$, all of which lie in $L$ and all of which are algebraic integers. Therefore $N(\beta)/\beta\in\mathcal{O}_L$. But $N(\beta)/\beta\in K$ and $\mathcal{O}_L\cap K=\mathcal{O}_K$ so that $N(\beta)/\beta\in\mathcal{O}_K$.