A Brief Introduction To Splitting Of Primes Over Number Fields

TL;DR

This paper explores the structure and analytic properties of Dedekind Zeta Functions over number fields, focusing on prime splitting in field extensions and their special cases, combining algebraic and analytic number theory insights.

Contribution

It provides a detailed analysis of the analytic behavior of Dedekind Zeta Functions and classifies primes that split completely in normal extensions of number fields.

Findings

Characterization of convergence and analyticity domains of Dedekind Zeta Functions

Classification of primes splitting completely in normal extensions

Analysis of special cases related to prime splitting

Abstract

The study of \textit{Dedekind Zeta Functions} over a number field extension uses different aspects of both \textit{Algebraic} and \textit{Analytic Number Theory}. In this paper, we shall learn about the structure and different analytic aspects of such functions, namely the domain of its convregence and analyticity at different points of $\mathbb{C}$ when the function is defined over any finite field extension $K$ over $\mathbb{Q}$ . Moreover, given any two Number Fields $L$ and $K$ over $\mathbb{Q}$ with $L$ being Normal over $K$, our intention is to classify and study the primes in $K$ which split completely in $L$. Also, we shall explore some special cases related to this result.