Tame An-extensions of

For every positive integer n and every finite set S of prime numbers, we construct An-extensions of unramified at all primes in S {∞}; moreover, these extensions are obtained as splitting fields of totally real monic polynomials in 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 .