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 $\mathbb {Q}$ of rational numbers in the field $\mathbb {C}$ of complex numbers. Let $G\sp {ab+\epsilon}$ be the quotient of $G$ universal for continuous homomorphisms to almost abelian profinite groups, and let $\mathbb {Q}\sp {ab+\epsilon}/\mathbb {Q}$ be the corresponding Galois extension. We prove that $\mathbb {Q}\sp {ab+\epsilon}$ 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 $\Gamma$-monomials by P. Das and by S. Seo.