A Density of Ramified Primes

TL;DR

This paper investigates the distribution of ramified primes in certain cyclic totally real number fields, providing explicit density formulas under a conjecture and unconditional results in the cubic case, based on prime ideal spin distributions.

Contribution

It introduces a family of number fields and derives explicit density formulas for ramified primes, advancing understanding of prime ramification in these fields.

Findings

Conditional density of primes with specific ramification behavior is between 0 and 1.

Unconditional results are obtained for cubic fields.

Density formulas are expressed explicitly in terms of the degree of the field.

Abstract

Let $K$ be a cyclic totally real number field of odd degree over $\mathbb{Q}$ with odd class number, such that every totally positive unit is the square of a unit, and such that $2$ is inert in $K/\mathbb{Q}$. We define a family of number fields $\{K(p)\}_p$, depending on $K$ and indexed by the rational primes $p$ that split completely in $K/\mathbb{Q}$, such that $p$ is always ramified in $K(p)$ of degree $2$. Conditional on a standard conjecture on short character sums, the density of such rational primes $p$ that exhibit one of two possible ramified factorizations in $K(p)/\mathbb{Q}$ is strictly between $0$ and $1$ and is given explicitly as a formula in terms of $[K:\mathbb{Q}]$. Our results are unconditional in the cubic case. Our proof relies on a detailed study of the joint distribution of spins of prime ideals.