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.