Trinomial extensions of with ramification conditions

This paper concerns trinomial extensions of with prescribed ramification behavior. We first characterize the positive integers n such that, for every finite set S of prime numbers, there exists a degree n monic trinomial in whose Galois group over is contained in the alternating group An and such that its discriminant is not divisible by any prime p in S. We also characterize the positive integers n such that, for a given finite set of primes S, there exist trinomial extensions with Galois group over contained in An which are not ramified at the primes of S. In addition, we study the existence of trinomial extensions of with Galois group An which are tamely ramified. In particular, we show that such extensions do exist for every odd n. On the other hand, we obtain that, for n≡4 (mod 8), every An-extension of defined by a degree n trinomial must be wildly ramified at p=2.