On the degree of extensions generated by finitely many algebraic numbers

Abstract A general criterion which may be viewed as a natural generalization of Eisenstein's criterion to a finite number of algebraic numbers over an arbitrary number field K is given. As a consequence of this criterion, sufficient conditions for the following general problems are obtained: Problem 1: Let x1, x2, …, xs be algebraic numbers with degrees ≤ n1, n2, …, ns over K, respectively. When does the field K(x1, x2, …, xs) have degree n1n2 … ns over K? Problem 2: Let x1, x2, …, xs be as stated in problem 1. Under what conditions is x1 + x2 + … + xs a primitive element of K(x1, x2, …, xs) over K? A series of interesting corollaries of our main results is also given. For example, we show that the algebraic number p 1 1 n 1 + p 2 1 n 2 + … + P s 1 n s has degree n1n2 … ns over Q, where p1, p2, …, ps are distinct primes.