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ALEÍCS A N D E R G R Y T C Z U K A N D J A R O S L A W G R Y T C Z U K
ON G E N E R A T O R S IN M U L T I P L I C A T I V E G R O U P OF T H E F I E L D Z p
A B S T R A C T : In the paper the following theorem is proved: "Let tl<
Zp be the multiplicative group of the field Z^, where p=2q+l and p , q are odd primes. Then 2, q+1, - 22 and - C q + 1 ) a r e generators in the group Z2 • p if p = 8 k + 3 and q, 2q-l, - q2 and — C 2 q - 1 )2 a r e g e n e r a t o r s in Z * if p = 8 k + 7 " .
P
This result is an extension of some earlier ones.
Baum [1] has g i v e n an i n t e r e s t i n g c r i t e r i a for c e r t a i n p r i m i t i v e roots. Vilansky t2J, using only the L e g e n d r e s y m b o l , proved t h e f o l l o w i n g r e s u l t : Let p and q a r e odd p r i m e s and p=2q+l. If q ^ l C m o d 4>, then q+1 is a p r i m i t i v e root m o d u l o p, w h i l e if q = 3 C m o d d), then q is a p r i m i t i v e root m o d u l o p.
In the present n o t e we g i v e s o m e e x t e n s i o n of t h i s r e s u l t p r o v i n g t h e f o l l o w i n g theorem:
THEOREM. Let Z * be the m u l t i p l i c a t i v e g r o u p of the field Zp , where p = 2 q + l and p,q are odd p r i m e s . Then 2, q+1, - 22 and
2 H<
- C q + 1 ) a r e g e n e r a t o r s in t h e g r o u p Zp if p = 8 k + 3 and q, 2 q - l ,
—q and - C 2 q - 1 ) ^ a r e g e n e r a t o r s in Z if p = 8 k + 7 . For the proof o f the T h e o r e m we nead two lemmas.
L E M M A 1. Let Z* be the m u l t i p l i c a t i v e g r o u p of the field Z
P P where p = 2 q + l and p,q are odd p r i m e s and let N Rp be the s e t o f
the q u a d r a t i c n o n — r e s i d u e s m o d u l o p. Then the set NR N.<2q> is
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t h e set of all g e n e r a t o r s o f t h e g r o u p 1(1
P R O O F OF L E M M A 1: Let R be the s e t o f all q u a d r a t i c r e s i d u e s
|f<
m o d u l o p. T h e n we h a v e t h a t R^ is a s u b g r o u p of Zp and s i n c e p=2q+l t h e g r o u p R h a s the o r d e r q. If b<£R t h e n b is n o t
P p a g e n e r a t o r in Z . S i n c e p
C2q> 2 = C p - l )q = C - l >q = —1Cmod p>,
h e n c e by E u l e r T h e o r e m w e get 2 q « N Rp. On the o t h e r hand we have
C 2 q ) 2 = C p - 1 )2 = 1Cmod p )
and t h e r e f o r e 2q h a s t h e o r d e r 2 and c a n n o t be a g e n e r a t o r in Zp. But the g r o u p Z^ h a s e x a c t l y
•p C p - 1 ) = <p C 2 q ) = q-1
g e n e r a t o r s and t h e r e f o r e the set. NRf>S(2q> is t h e set of all g e n e r a t o r s in Z*.
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L E M M A 2. Let g be a g e n e r a t o r o f t h e g r o u p Z w h e r e p = 4 k + 3 . T h e n - g is a l s o a g e n e r a t o r in Z^.
P R O O F OF L E M M A 2: Let p = 4 k + 3 and g be a g e n e r a t o r in Z*. T h e n we have
Z * = { gk ; k = l , 2 , . . . .p-1 = 4 k + 2 } . By Euler t h e o r e m we h a v e
g 2 = -1 and t h e r e f o r e g2 k*1 — -1- From last e q u a l i t y we g e t
C 2.1> - g2 = g2 k*3
It is easy to s e e t h a t C2k+3, 4 k + 2 = p - l > = l and t h e r e f o r e g2 k + 3 is a g e n e r a t o r in Z * t h u s by C 2 . 1 ) L e m m a 2 f o l l o w s . P R O O F OF T H E T H E O R E M . F i r s t we r e m a r k that if p=2q+l w h e r e p,q a r e odd p r i m e s t h e n p = 4 m + 3 , b e c a u s e f o r p=4m+l t h e e q u a l i t y p = 2 q + i is i m p o s i b l e . H e n c e p = 8 k + 3 o r p=8k+7. If
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p = 8 k + 3 then -1 and the n u m b e r 2 is a q u a d r a t i c n o n — r e s i d u e m o d u l o p. F r o m L e m m a i we get t h a t 2 is a g e n e r a t o r in Zp. On t h e o t h e r hand we h a v e
2 C q + l ) = 2 q + 2 = 2q+l+l = p+1 = lCmod p>
and t h e r e f o r e q+1 is a g e n e r a t o r in Z^ a s t h e i n v e r s e e l e m e n t with r e s p e c t to 2.
Let p = 8 k + 7 , t h e n 2 is not a g e n e r a t o r in Z * . S i n c e t h e r e a r e e x a c t l y £>Cp-l) g e n e r a t o r s in Zp and t h e e l e m e n t 1 i s not a g e n e r a t o r t h u s t h e r e e x i s t s at least o n e
>ii n u m b e r g such that C g , p — 1 ) > 1 and g is a g e n e r a t o r in Z^.
B e c a u s e p=2q+l t h u s q|p-l and C q , p - l ) > l and 2 is not a
#
g e n e r a t o r t h u s the n u m b e r q must b e a g e n e r a t o r in Zp. V e have
qC2q-l> = q C 2 q + l - 2 ) = q C p - 2 ) s -2q = 1 - p = lCmod p). tfi and so 2q—1 is a g e n e r a t o r in Z^. T h e last part o f a s s e r t i o n f o l l o w s from L e m m a 2 and the p r o o f is c o m p l e t e .
R E F E R E N C E S
111 J.D. Baum, A note on p r i m i t i v e roots, M a t h , M a g . 3 8 C1965> 12-14.
12 Í A. W i l a n s k y , P r i m i t i v e r o o t s w i t h o u t q u a d r a t i c r e c i p r o c i t y , Math. Mag. 49 ( 1 9 7 6 > , 146.
