  Let k be a number field, and denote by k^[d] the compositum of all degree d
extensions of k in a fixed algebraic closure. We first consider the question of
whether all algebraic extensions of k of degree less than d lie in k^[d]. We
show that this occurs if and only if d < 5. Secondly, we consider the question
of whether there exists a constant c such that if K/k is a finite subextension
of k^[d], then K is generated over k by elements of degree at most c. This was
previously considered by Checcoli. We show that such a constant exists if and
only if d < 3. This question becomes more interesting when one restricts
attention to Galois extensions K/k. In this setting, we derive certain
divisibility conditions on d under which such a constant does not exist. If d
is prime, we prove that all finite Galois subextensions of k^[d] are generated
over k by elements of degree at most d.
