Tame An-extensions of Q

For every positive integer n and every finite set S of prime numbers, we construct An-extensions of Q unramified at all primes in S∪{∞}; moreover, these extensions are obtained as splitting fields of totally real monic polynomials in Z [X] of degree n whose discriminant is not divisible by any prime number p in S. As a corollary, we obtain that there exist infinitely many linearly disjoint tamely ramified An-extensions of Q .