  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.
