# Closest cyclotomic integer to a cyclotomic number?

Let's take a cyclotomic field of the form $K=\mathbb{Q}(\zeta_n)$ where $\zeta_p$ is the $n$th root of unity. Then the ring of integers of $K$ is $\mathcal{O}_K= \mathbb{Z}(\zeta_n)$. Is there a generalisation of the rounding function $\left \lfloor \cdot \right \rceil: \mathbb{Q} \to \mathbb{Z}$ to some rounding function $\left \lfloor \cdot \right \rceil_K : K \to \mathcal{O}_K$ for cyclotomic fields that rounds a cyclotomic number to its "nearest" cyclotomic integer?

EDIT: I found something that might be useful. The following definition comes from https://hal.archives-ouvertes.fr/hal-00632997v1/document:

Definition: For any $\eta \in K$, the real number $m_K(\eta)= \min_{z \in \mathcal{O}_K}|N_{K/\mathbb{Q}}(\eta - z)|$ is the Euclidean minimum of $\eta$.

Does this give us a generalisation of the rounding function, and if so does the "rounding" function only hold for Euclidean domains?

• Also I was reading that each ring of integers is isomorphic to the $n$-dimensional integer lattice $\mathbb{Z}^n$ - would finding the closest point on such a lattice correspond to finding the closest algebraic integer? Jun 13 '18 at 9:56
• Consider When is $\mathbb{Z}[\alpha]$ dense in $\mathbb{C}$ and e.g. $\mathbb{Z}[\zeta_8]$. With the usual distance, there is no nearest algebraic integer. Jun 13 '18 at 12:18
• If $\mathbb{Z}[\zeta_n]$ is dense in $\mathbb{C}$, then there are infinitely many integers from $\mathbb{Z}[\zeta_n]$ in every neighborhood of a given non-integer element of $\mathbb{Q}[\zeta_n]$ (with the continuous distance). Therefore there is no nearest integer then. Jun 14 '18 at 10:30
• You can certainly convert this into a reasonably related nearest lattice point prolem. Just use the usual embedding of $K=\Bbb{Q}(\zeta_n)$ into, not just $\Bbb{C}$ but $\Bbb{C}^{\phi(n)/2}$. If $z\in K$, then its image is the vector $(\sigma_i(z))_{1\le i\le\phi(n)/2}$ where $\sigma_i$ ranges over a set of cosets of the subgroup generated by the usual complex conjugation in the Galois group. This is bread-and-butter in ANT. Then $\mathcal{O}_K$ is turned into a discrete lattice, and by blowing up the dimension we avoid the topological difficulties ccorn commented on. Jun 15 '18 at 6:31
• Related MO thread: Find closest integers in Euclidean rings Jun 16 '18 at 7:51

Short answer: Yes, there is such a function. Just map every element of your number field to $0$. Since you do not specify what properties your rounding function should have, you can't even complain that this answer is trivial.
As suggested by the remarks, you can choose an integral basis and round each element of the number field to the integer that is coordinatewise closest to this number. This also gives you many rounding functions, and you if you choose $1$ as an element of your integral basis then you may even pick a rounding function that restricts to the usual rounding function in the rationals.
The Euclidean minimum has little to do with the question; its image is a real number, most often rational, but rarely integral. You can of course map $\eta$ to some $z$ that minimzes the absolute value of the norm of $\eta - z$, but this is not well defined if your number field has nontrivial units.