Splitting primes

TL;DR

This paper provides an elementary overview of the law governing primes where a polynomial splits completely, illustrating key results from the Langlands programme and recent breakthroughs like modularity of elliptic curves and Serre's conjecture.

Contribution

It offers an accessible introduction to complex concepts in number theory and highlights recent significant results through illustrative examples.

Findings

Illustrates the law governing primes for polynomial splitting

Shows examples of modularity of elliptic curves

Explains proof of Serre's conjecture

Abstract

We give an elementary introduction, through illustrative examples but without proofs, to one of the basic consequences of the Langlands programme, namely the law governing the primes modulo which a given irreducible integral polynomial splits completely. Some recent results, such as the modularity of elliptic curves over the rationals, or the proof of Serre's conjecture by Khare and Wintenberger, are also illustrated through examples.