Prime decomposition in the anti-cyclotomic extension

For an imaginary quadratic number field $K$ and an odd prime number $l$, the anti-cyclotomic $\mathbb{Z}_l$-extension of $K$ is defined. For primes $\mathfrak{p}$ of $K$, decomposition laws for $\mathfrak{p}$ 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 $\mathbb{Z}_l$-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.