Differential principal factors and Polya property of pure metacyclic fields

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.