Let $K$ be a number field of degree $n$ and $\mathfrak{a}$ be an ideal in its ring of integers $\mathcal{O}_K$. We can consider:
- The norm $N(\mathfrak{a})$ of $\mathfrak{a}$.
- The norms $N(x)$ of the elements $x\in\mathfrak{a}$.
It is well known that:
- For all $x\in\mathfrak{a}$, $N(\mathfrak{a})|N(x)$, so $\lvert N(x) \rvert \ge N(\mathfrak{a})$
- $N(\mathfrak{a})\in\mathfrak{a}$
- By point 2., $\mathfrak{a}$ contains an element of norm $N(\mathfrak{a})^n$.
But does there exist an element $x\in\mathfrak{a}$ such that precisely $\lvert N(x) \rvert =N(\mathfrak{a})\ ?$