let $k$ be a number field. given a positive integer $n$, does there exist an algebraic extension $l/k$ with local degree $n$ at all finite places of $k$, and degree two at the real places if $n$ is even? this problem comes from brauer groups of fields: given a field $k$ and a positive integer $n$, is there an algebraic extension $l/k$ such that the relative brauer group $br(l/k)$ is equal to the $n$-torsion subgroup of the brauer group $br(k)$ of $k$? in general the answer to the latter question is no, a counterexample coming from two dimensional local fields. the first problem is essentially equivalent to the second when $k$ is a number field, in which case no counterexample has been found as yet. in fact, the answer is affirmative when $k$ is the rationals $\\bbb q$, and for general global fields under certain hypotheses.