The Galois theory of the lemniscate

TL;DR

This paper investigates the Galois groups associated with division points of the lemniscate, employing class field theory and polynomial irreducibility to deepen understanding of their algebraic structure.

Contribution

It introduces lemnatomic polynomials as analogs of cyclotomic polynomials and computes Galois groups via two distinct methods, expanding classical Galois theory to elliptic functions.

Findings

Computed Galois groups using class field theory

Proved irreducibility of lemnatomic polynomials

Connected lemnatomic polynomials to Chebyshev polynomials

Abstract

This article studies the Galois groups that arise from division points of the lemniscate. We compute these Galois groups two ways: first, by class field theory, and second, by proving the irreducibility of lemnatomic polynomials, which are analogs of cyclotomic polynomials. We also discuss Abel's theorem on the lemniscate and explain how lemnatomic polynomials relate to Chebyshev polynomials.