A Ring of Pythagorean Triples over Quadratic Fields

Abstract

Let K be a quadratic field and let R be the ring of integers of K such that R is a unique factorization domain. The set P of all Pythagorean triples in R is partitioned into P _η , sets of triples 〈α, β, γ〉 in P where η = γ − β. We show the ring structures of each P _η and P from the ring structure of R.