Prime decomposition in the anti-cyclotomic extension

Abstract

For an imaginary quadratic number field K and an odd prime number l, the anti-cyclotomic Z l -extension of K is defined. For primes p of K, decomposition laws for p in the anti-cyclotomic extension are given. We show how these laws can be applied to determine if the Hilbert class field (or part of it) of K is Z l -embeddable. For some K and l, we find explicit polynomials whose roots generate the first step of the anti-cyclotomic extension and show how the prime decomposition laws give nice results on the splitting of these polyniomials modulo p. The article contains many numerical examples.