# A Note on the equivalence of the parity of class numbers and the signature ranks of units in cyclotomic fields

## TL;DR

This paper explores the relationship between class number parities and signature ranks of units in prime power cyclotomic fields, correcting previous misconceptions with a specific counterexample.

## Contribution

It clarifies the equivalence of class number and signature rank parities and provides a counterexample to prior assumptions in cyclotomic field theory.

## Key findings

- Counterexample showing signature rank of units differs from circular units
- Correction of previous literature misstatements
- Insights into parity relationships in cyclotomic fields

## Abstract

We collect some statements regarding equivalence of the parities of various class numbers and signature ranks of units in prime power cyclotomic fields. We correct some misstatements in the literature regarding these parities by providing an example of a prime cyclotomic field where the signature rank of the units and the signature rank of the circular units are not equal.