Kronecker-Weber plus epsilon
Abstract
Description
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 $G^{\ab+ε}$ be the quotient of $G$ universal for homomorphisms to almost abelian profinite groups and let $\QQ^{\ab+ε}/\QQ$ be the corresponding Galois extension. We prove that $\QQ^{\ab+ε}$ 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 $Γ$-monomials by P.~Das and by S.~Seo.
This paper has appeared as Duke Math. J. 114 (2002) 439-475.