Pythagorean triples over number filelds

Abstract

It is well-known that each primitive Pythagorean triple of natural numbers is uniquely determined by a pair of natural numbers which are relatively prime and have different parities. In addition, Pythagorean triples were also characterized in an arbitrary unique factorization domain. The set of Pythagorean triples of integers was also studied in terms of its structure. Binary operations can be defined so that this set is a semigroup, a group or a ring. In this thesis, we extend the ideas and investigate properties and structures of the semigroup of Pythagorean triples over Gaussian integers, the ring of Pythagorean triples over quadratic fields and biquadratic fields, and the group of Pythagorean triples over any number fields. Moreover, we determine all Pythagorean triples in the ring of integers of any number field.