In this paper by exploiting the properties of transitive permutation groups of prime degree we provide an answer to the following problem: given two number fields K and L both of prime degree p over Q, what values the degree of their compositum KL can take? We show that if K and L are linearly disjoint over Q, then necessarily KL has degree p² or, for example, if K and L are number fields of prime degree p such that p=(qⁿ-1)/(q-1) with q prime or a power of a prime, n>2, and some intermediate group between the projective special linear group PSL(n,q) and the projective semilinear group PΓL(n,q) is realizable over Q, then the degree of KL is pq^n-1. In addition, for any divisor s of p-1, there exist number fields K and L of prime degree p such that their compositum KL has degree ps. As a numerical application, we determine the complete list of values the degree of compositum KL can take if K and L are two number fields of degree 13. We also give an answer to the related problem, namely, given two algebraic numbers α and β both of prime degree p, what values the degree of α+β and αβ can take?

The author is grateful to A. Dubickas for pointing out many helpful suggestions, and to
P. Drungilas for the proofs of Lemma 2.4 and Lemma 2.8. The author also thanks the referee for valuable comments.