Kronecker-Weber plus epsilon

We say that a group is {\em 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 $\QQ$ of rational numbers in the field of complex numbers. Let $G^{\ab+\epsilon}$ be the quotient of $G$ universal for homomorphisms to almost abelian profinite groups and let $\QQ^{\ab+\epsilon}/\QQ$ be the corresponding Galois extension. We prove that $\QQ^{\ab+\epsilon}$ is generated by the roots of unity, the fourth roots of the (rational) prime numbers and the square roots of certain sine-monomials. The inspiration for the paper came from recent studies of algebraic $\Gamma$-monomials by P.~Das and by S.~Seo. This paper has appeared as Duke Math. J. 114 (2002) 439-475.