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Unformatted text preview: A WORD ABOUT PRIMITIVE ROOTS PETE L. CLARK Let N be a positive integer. An integer g is said to be a primitive root modulo N if every element x of ( Z /N Z ) × is of the form g i for some positive integer i . Equivalently, the finite group ( Z /N Z ) × is cyclic and g (mod N ) is a generator. We’d like to find primitive roots mod N , if possible. There are really two problems: Question 1. For which N does there exist a primitive root modulo N ? Question 2. Assuming there does exist a primitive root modulo N , how do we find one? How do we find all of them? We can and shall give a complete answer to Question 1. We already know that the group of units of a finite field is finite, and we know that Z /N Z is a field if (and only if) N is prime. Thus primitive roots exist modulo N when N is prime. When N is not prime we might as well ask a more general question: what is the structure of the unit group ( Z /N Z ) × ? From our work on the Chinese Remainder theorem, we know that if N = p a 1 1 ··· p a r r , there is an isomorphism of unit groups ( Z /N Z ) × = Z / ( p a 1 1 ··· p a r r Z ) × ∼ = r Y i =1 ( Z /p a i i Z ) × . Thus it is enough to figure out the group structure when N = p a is a prime power. Theorem 1. The finite abelian group ( Z /p a Z ) × is cyclic whenever p is an odd prime, or when p = 2 and a is 1 or 2 . For a ≥ 3 , we have ( Z / 2 a Z ) × ∼ = Z 2 × Z 2 a- 2 . Before proving Theorem 1, let us nail down the answer it gives to Question 1. Corollary 2. Primitive roots exist modulo N in precisely the following cases: (i) N = 1 , 2 or 4 ....
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This note was uploaded on 10/26/2011 for the course MATH 8410 taught by Professor Staff during the Fall '11 term at UGA.
- Fall '11
- Number Theory