The Galois theory of the lemniscate

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.Comment: The revised version adds four references and some historical remarks. We also note that a special case of Theorem 4.1 appears in Lemmermeyer's Reciprocity Law