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.