Cyclotomic splitting fields

Suppose $k$ is an algebraic number field and $D$ a finite-dimensional central division algebra over $k$. It is well known that $D$ has infinitely many maximal subfields which are cyclic extensions of $k$. From the point of view of group representations, however, the natural splitting fields are the cyclotomic ones. Accordingly it has been conjectured that $D$ must have a cyclotomic splitting field which contains a maximal subfield. The aim of this paper is to show that the conjucture is false; we will construct a counter-example of exponent $p$, one for every prime $p$.