Classification of ideals of algebras of real matrices and its applications in the number theory

Expicit classifications of ideals of algebras of matrices of order two over the field of real numbers, and over the field of rational numbers and over finite fields are given. The find classification allows to calculate the amount of primitive Pythagorean triples and the numbers of points of discrete circles in finite fields. Proofs are based on the isomorphism between algebra of matrices of order two over the field of real numbers and a generalized quaternion algebra which was studied in previous work by the author.