A degree problem for the compositum of two number fields*

The triplet (a, b, c) of positive integers is said to be compositum-feasible if there exist number fields K and L of degrees a and b, respectively, such that the degree of their compositum KL equals c. We determine all compositum-feasible triplets (a, b, c) satisfying a ≤ b and b ∈ {8, 9}. This extends the classification of compositum-feasible triplets started by Drungilas, Dubickas, and Smyth [5]. Moreover, we obtain several results related to triplets of the form (a, a, c). In particular, we prove that the triplet (n, n, n(n ? 2)) is not compositum-feasible, provided that n ≥ 5 is an odd integer.