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

Hence by (2.21) we have (2.23) p

2 - N 2

a v 2

a From (2.23) it follows that the equation \x

2-dy

2\=4

ap has a solution in integers, x,y and proof of Lemma 2 is complete. Result We can prove the following Theorem. Let K = Q{Jd), d>0, d ^Z

s\L

d. Then the ring R

k of K is the ring with nonuniqueness of factorization. Proof. Suppose that for some 0 < d e Z

s \ L

d the ring R

k is the ring with uniqueness of factorization. By Lemma 1 it follows that there are odd primes p,q,q* such that p\d, q\d and (3.1.) <1 J — I <Pj -1 V) I P j = -1 From (3.1) and Lemma 2 it follows that the equation (3.2) \x

2 -dy

2\~ 4

aq* has a solution in integers x,y. From (3.2) we have (3.3) Jt

2 -dy

2 =4

aq or jc

2 - dy

2 = -4

aq\ From (3.3) it follows that for p\d and q\d we have (3.4.) r 4

aq ^ v P = 1 or '-4 V^ = 1. But on the other hand from (3.1) we obtain 89