Number Theory - Primitive Roots

[mattmns]

Here is the question from the book:
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Determine a primitive root modulo 19, and use it to find all the primitive roots.
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$$\varphi(19)= 18$$

And 18 is the order of 2 modulo 19, so 2 is a primitive root modulo 19, but I am not sure of how to use that to find all primitive roots modulo 19. My only idea is that we need to find what values of g satisfy $g^{18} \equiv 1 \ \text{mod 19}$. However, I am not sure how to solve that equation. Any ideas? Thanks!

 

[Hurkyl]

Well, you know that the unit group of Z/19Z is simply a cyclic group of order 18, right?

If that doesn't help, don't forget that g is a power of 2. Now, you know that everything in Z/19Z satisfies g^18 = 1... the things you're interested in are the things that do not also satisfy g^9 = 1 or g^6 = 1. (Do you see why?)

 

[mattmns]

Unfortunately I don't know much algebra, and our number theory class has not focused on the algebra behind it, so I don't really understand what you are saying.

 

[Hurkyl]

Z/19Z is simply the residue classes modulo 19.

 

[mattmns]

Thanks, now I see the idea behind it, and I see how to find the others. Seems kind of obvious now

 

[howdowedothis]

why are we interested in the things that DO not satisfy g^9 = 1 or g^6 = 1?