Non-definability of rings of integers in most algebraic fields

TL;DR

This paper proves that in most algebraic extensions of the rational numbers, the ring of integers cannot be explicitly defined within the field, indicating a widespread non-definability phenomenon.

Contribution

It demonstrates that the collection of algebraic fields where the ring of integers is definable is meager, highlighting the rarity of such definability.

Findings

The set of fields with definable rings of integers is meager.

Most algebraic extensions do not allow definability of their rings of integers.

The result applies to both $ extbf{Z}$ and $ extbf{O}_F$ in algebraic extensions.

Abstract

We show that the set of algebraic extensions $F$ of $\mathbb{Q}$ in which $\mathbb{Z}$ or the ring of integers $\mathcal{O}_F$ are definable is meager in the set of all algebraic extensions.