The number of ramified primes in number fields of small degree

TL;DR

This paper studies how many primes ramify in number fields of degree up to 5, showing that their distribution follows a normal distribution similar to the Erdős-Kac theorem, with mean and variance proportional to log log X.

Contribution

It extends the Erdős-Kac type normal distribution result to ramified primes in degree d <= 5 number fields, providing new probabilistic insights.

Findings

Number of ramified primes in degree d <= 5 fields is normally distributed.

Mean and variance of the distribution are proportional to log log X.

Results generalize classical probabilistic number theory to algebraic number fields.

Abstract

In this paper we investigate the distribution of the number of primes which ramify in number fields of degree d <= 5. In analogy with the classical Erdos-Kac theorem, we prove for S_d-extensions that the number of such primes is normally distributed with mean and variance log log X.