Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1991. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 20)

Aleksander Grytczuk and Jaroslaw Grytczuk: On generators in multiplicative group of the Fieid Zp

- 39 ALEÍCS ANDER GRYTCZUK AND JAROSLAW GRYTCZUK ON GENERATORS IN MULTIPLICATIVE GROUP OF THE FIELD Z p ABSTRACT: In the paper the following theorem is proved: "Let tl

< Z

p be the multiplicative group of the field Z^, where p=2q+l and p,q are odd primes. Then 2, q+1

, -2

2 and 2 • -Cq+1) are generators in the group Z

p if p=8k+3 and q, 2q-l

, -q

2 and —C2q-1)

2 are generators in Z* if p=8k+7". P This result is an extension of some earlier ones. Baum [1] has given an interesting criteria for certain primitive roots. Vilansky t2J, using only the Legendre symbol, proved the following result: Let p and q are odd primes and p=2q+l

. If q^lCmod 4>, then q+1 is a primitive root modulo p, while if q=3Cmod d

), then q is a primitive root modulo p. In the present note we give some extension of this result proving the following theorem: THEOREM. Let Z* be the multiplicative group of the field Z

p , where p=2q+l and p,q are odd primes. Then 2, q+1

, -2

2 and 2 H< -Cq+1) are generators in the group Z

p if p=8k+3 and q, 2q-l

, —q and -C2q-1)^ are generators in Z if p=8k+7. For the proof of the Theorem we nead two lemmas. LEMMA 1. Let Z* be the multiplicative group of the field Z P P where p=2q+l and p,q are odd primes and let NR

p be the set of the quadratic non—residues modulo p. Then the set NR N.<2q> is