A simple criterion for the class number of a quadratic number field to be one

Abstract

Let [Formula: see text] be a square-free positive integer, [Formula: see text] if [Formula: see text] and [Formula: see text] otherwise. Let [Formula: see text] and [Formula: see text] be integers, where [Formula: see text] is a prime. Suppose that [Formula: see text] for some integer [Formula: see text]. Suppose that there exist integers [Formula: see text] and [Formula: see text] such that [Formula: see text]. We prove that if [Formula: see text] is [Formula: see text] or a prime for all integers [Formula: see text] with [Formula: see text], then the class number of the field [Formula: see text] is [Formula: see text].