On a Problem of Hasse

A p-adic method to construct explicitly a generating automorphism of the Hilbert classfield over Q(V-47) and to perform Tshirnhausen transformations for generating equations of the real subfield is developed. I. Let f(x) be a monic polynomial with coefficients in Z, irreducible of degree n over Q, with a a real root, let k=Q(-\d), d<O E= Q(0), K = E(V\d) and K is normal over Q and cyclic of degree n over k. How to find a generating element of G(K/k), where G(K/k) is the Galois group of K over k? Here we give a p-adic method to construct such an automorphism. In the end, we shall give some examples to demonstrate our method. By a theorem given in (2) there are infinitely many rational prime numbers p which decompose in k into the product of two distinct prime ideals which stay in- decomposed in K. Those are the ones with decomposition group equal to G(K/k) and not dividing the discriminant of K over Q. Among them there is one which does not even divide the characteristic b of the factor module of R K over )E )k. (We denote by OF the ring of the algebraic integers of the algebraic number field F.) Let p = 1lP2 in k, pi 7 P2 are prime ideals in k and let Hi = DPiK, i = 1, 2, 93i prime ideals in kDK. Since k is imaginary quadratic the two prime ideals ,i, P2 are