Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1993. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 21)

A. Grytczuk and J. Kacierzynski: On Factorization in real quadratic number fields

In this paper we prove by simple elementary method without using class-field theory the following. Theorem. Let Z

s denote the set of all square-free integers and L

d - {d.d - p,2qor qr,q = r = 3(mod4)} and let K = Q( Jd)

, d > 0 and d e Z

s \ L

d

. Tlien the ring R

K of K is the ring with nonuniqueness of factorization. It is easy to see that from our Theorem follows also the corollary which follows from Herz's result Let Z

s denote the set of all squarefree positive integers and (2.1) L

d={d = p\2q~ or qr,q'= r'= 3(mod 4),pq\r are primes} Then we can prove the following Lemma 1. For every d zZ

s\L

d there exist the odd primes p,q,q* suchthat (2.2) p\d,q\d (maybe p = q) and (2.3.) <7 ) = 1, í * V <i \ p = -i. ProoL Since d eZ

s\ L

d thus it suffice to consider the following four cases: 1° d - 2p, p = 1 (mod 4) is a prime. 82