Kronecker-Weber plus epsilon

Abstract

We say that a group is almost abelian if every commutator is central and squares to the identity. Now let G be the Galois group of the algebraic closure of the field ℚ of rational numbers in the field ℂ of complex numbers. Let Gab+ε be the quotient of G universal for continuous homomorphisms to almost abelian profinite groups, and let ℚab+ε/ℚ be the corresponding Galois extension. We prove that ℚab+ε is generated by the roots of unity, the fourth roots of the rational primes, and the square roots of certain algebraic sine-monomials. The inspiration for the paper came from recent studies of algebraic Γ-monomials by P Das and by S. Seo.