Making sense of capitulation: reciprocal primes (Q2804245)

Let \(\ell\) be a prime number, let \(K\) be a number field containing a primitive \(\ell\)-th root of unity and let \(\mathfrak{p}\) be a prime ideal of the ring of integers \(\mathcal{O}_K\) of \(K\) which belongs to an ideal class of order \(\ell\). The problem studied in the paper under review is to find some criteria to decide whether \(\mathfrak{p}\) capitulates (i.e., becomes principal) in an abelian extension \(L/K\). To this aim the author defines \(F_K=K(\sqrt[\ell]{\mathcal{O}_K^\times})\) and fixes a generator \(a_{\mathfrak{p}}\) of the principal ideal \(\mathfrak{p}^\ell\). Furthermore, he calls a prime ideal \(\mathfrak{q}\) of \(\mathcal{O}_K\) a \textit{reciprocal} of \(\mathfrak{p}\) if \(\mathfrak{q}\) does not divide \(\ell\) and its Frobenius element in \(\mathrm{Gal}(F_K(\sqrt[\ell]a_{\mathfrak{p}})/K)\) generates the subgroup \(\mathrm{Gal}(F_K(\sqrt[\ell]a_{\mathfrak{p}})/F_K)\). The main result of the paper is that \(\mathfrak{p}\) capitulates in \(L\) if and only if \textit{every} reciprocal prime \(\mathfrak{q}\) of \(\mathfrak{p}\) is not a norm in a suitable ray class group. The main theorem is then followed by a corollary, which is used in the last part of the paper to do some computations in specific examples.NEWLINENEWLINERemark of the reviewer: it does not appear to be clear how part (a) of the corollary should follow from the theorem since the conditions in the corollary are given just for \textit{one} of the reciprocal primes of \(\mathfrak{p}\), not for \textit{everyone} as would be required to apply the theorem.