# Differential principal factors and Polya property of pure metacyclic fields

## TL;DR

This paper generalizes principal factorization theory from pure cubic to pure quintic and metacyclic fields, classifies their types, and investigates the Polya property using Galois cohomology, supported by extensive numerical analysis.

## Contribution

It extends Barrucand and Cohn's principal factorization classification to pure quintic and metacyclic fields, establishing new criteria for the Polya property and providing detailed group structure analysis.

## Key findings

- 13 possible types of pure metacyclic fields identified
- Pure metacyclic fields of only 1 type are not Polya fields
- Numerical verification conducted for 900 radicands in range 2 to 999

## 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.