Pell-type equations and class number of the maximal real subfield of a cyclotomic field

TL;DR

This paper studies the solvability of specific Pell-type equations and uses these results to identify families of maximal real subfields of cyclotomic fields with class number greater than one.

Contribution

It provides new criteria for solving Pell-type equations and applies these to determine class numbers of certain cyclotomic subfields.

Findings

Identified conditions under which Pell-type equations are solvable.

Constructed families of cyclotomic subfields with class number > 1.

Extended understanding of class number behavior in cyclotomic fields.

Abstract

We investigate the solvability of the Diophantine equation $x^2-my^2=\pm p$ in integers for certain integer $m$ and prime $p$. Then we apply these results to produce family of maximal real subfield of a cyclotomic field whose class number is strictly bigger than $1$.