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

Consider the case 3°. Let \PJ = +1 and \Pi = -1 for / = 2,3,...,£. Since (p

i

ip

J-) = 1 for / * j thus by Chinese remainder theorem we obtain that there exists a positive integer u such that u = r (mod p

i

)

, for / = 1,2,..., k. Similarly as in the case 2°, let m

0 denote the number statisfying p

x...

p

km

0 + u = 3 (mod 8) and let rm = 8/V--AW + ta ...p

km

0 + u). Thus we obtain for some m, q=r

m and q = 3 (mod 8). Therefore we obtain * \ (2.17) Í ypj ( u .A IV vA = 1 and KPi) Pij t-' for / = 2,3,...,*. By (2.17) and reciprocity law of Gauss it ( „ \ = -1, <1 ) follows that A \<i = 1 for # = 2,3,..., k and KHTherefore we obtain fd\ * = 1, ( -q = -1 , U J V A > 1, and the case 3° is proved. For the proof the case 4 we suppose that n f r \ ( r \ 1 2 V Aj va ; = 1 and = -1 for /' = Since (/?,p.

) = 1 for thus by Chinese remainder theorem we obtain that there exists a positive integer u such 86