Primitive roots?

Can a number have more than 1 primitive root?

Thanks

 

From the name, I get the impression that there should be only one primitive root for each number but there are more due to the definition that a primitive root exists when the primitive root to the power of phi(n) has a remainder of 1 when divided by n.

Finding the primitive of a number, n (if it exists) is quiet a long process of trial and error is it not?

 

What do you mean by relatively prime?

How many numbers do have to try to ensure this number has no primitive roots?

Thanks Hurkyl

 

Also, prime numbers always have at least one primitive root but do they have infinitely many?

Are one primitive root usually prefered to another?

 

One excercise was find the primitive root of 12.

I did it by trying to find a number 'a' where the ord of the number 'a' to the base 12 equal phi(12)

phi(12) = 2

and so I found 5^2 is congruent to 1 (mod 12). The power of 5, that is 2 is also the smallest power for which 5^i is congruent to 1 (mod 12).

the gcd(5,12)=1

Hence i=phi(12)=ord5(mod12) which matches the definition of primitive root. 'a' or the primitive root therefore is 5.



But the back of the book said that 12 has no primitive root.

Is the book wrong or am I wrong?

Thanks