Differential principal factors and Polya property of pure metacyclic fields

Abstract

Barrucand and Cohn's theory of principal factorizations in pure cubic fields \(\mathbb{Q}(\sqrt[3]{D})\) and their Galois closures \(\mathbb{Q}(\zeta_3,\sqrt[3]{D})\) with \(3\) types is generalized to pure quintic fields \(L=\mathbb{Q}(\sqrt[5]{D})\) and pure metacyclic fields \(N=\mathbb{Q}(\zeta_5,\sqrt[5]{D})\) with \(13\) possible types. The classification is based on the Galois cohomology of the unit group \(U_N\), viewed as a module over the automorphism group \(\mathrm{Gal}(N/K)\) of \(N\) over the cyclotomic field \(K=\mathbb{Q}(\zeta_5)\), by making use of theorems by Hasse and Iwasawa on the Herbrand quotient of the unit norm index \((U_K:N_{N/K}(U_N))\) by the number \(\#(\mathcal{P}_{N/K}/\mathcal{P}_K)\) of primitive ambiguous principal ideals, which can be interpreted as principal factors of the different \(\mathfrak{D}_{N/K}\). The precise structure of the group of differential principal factors is determined with the aid of kernels of norm homomorphisms and central orthogonal idempotents. A connection with integral representation theory is established via class number relations by Parry and Walter involving the index of subfield units \((U_N:U_0)\). Generalizing criteria for the Polya property of Galois closures \(\mathbb{Q}(\zeta_3,\sqrt[3]{D})\) of pure cubic fields \(\mathbb{Q}(\sqrt[3]{D})\) by Leriche and Zantema, we prove that pure metacyclic fields \(N=\mathbb{Q}(\zeta_5,\sqrt[5]{D})\) of only \(1\) type cannot be Polya fields. All theoretical results are underpinned by extensive numerical verifications of the \(13\) possible types and their statistical distribution in the range \(2\le D<10^3\) of \(900\) normalized radicands.