## 10 An application of elliptic functions in algebra – solution of the general quintic equation

### Introduction

One of the earliest applications of the theory of elliptic functions, indeed one of its origins, is to be found in ideas arising from Gauss’ criterion for the constructibility by ruler and compass of a regular polygon of n sides, namely that n = 2^{m}p_{1}p_{2} . . .p_{r}, where the p_{i} are distinct Fermat primes, of the form (see Gauss, 1801, Section 7, or Hardy & Wright, 1979, Chapter V). That problem played a significant part in the development of Galois theory and also in the work of Abel, who proved that, for those values of n given by Gauss, it is ...