On the number of ramified primes in specializations of function fields over $\mathbb{Q}$

TL;DR

This paper proves a central limit theorem for the distribution of ramified primes in specialized Galois extensions of the rationals, providing new insights into their statistical behavior and Galois-theoretic applications.

Contribution

It establishes a central limit theorem for the number of ramified primes in specialized Galois extensions, a novel statistical result in number theory.

Findings

Central limit theorem for ramified primes in specializations

Quantitative description of ramification distribution

Galois theoretical applications derived from the results

Abstract

We study the number of ramified prime numbers in finite Galois extensions of $\mathbb{Q}$ obtained by specializing a finite Galois extension of $\mathbb{Q}(T)$. Our main result is a central limit theorem for this number. We also give some Galois theoretical applications.