# Coarse cohomology types of pure meta cyclic fields

## TL;DR

This paper investigates the cohomological properties of certain pure septic fields, revealing unexpected behaviors in their class numbers and factorizations through direct computational methods.

## Contribution

It provides new insights into the cohomology types of pure meta cyclic fields, especially regarding their class numbers and differential factorizations.

## Key findings

- Pure septic fields with specific prime radicands exhibit unique cohomological behavior.
- Class numbers of these fields are not divisible by 7 despite containing differential principal factorizations.
- Direct computation confirms the unexpected properties of these fields.

## Abstract

An unexpected behavior of pure septic fields L = Q(D^1/7) with certain prime radicands D congruent to 2 or 4 modulo 7, which split in the cyclotomic field of seventh roots of unity, is proved by direct computation. Whereas the Galois closures N = Q(zeta_7, D^1/7) of these fields contain relative differential principal factorizations in the kernel of the norm of N/L, the class numbers of L and N are not divisible by 7.