  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.
