Find the number of primitive roots in , .

I cannot find an example from my lecture notes about primitive roots in . What is the definition of primitive roots in ?

The answers are 40 and none respectively.

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- Apr 25th 2012, 07:23 AMmath2011Find the number of primitive roots in Z_101, Z_12
Find the number of primitive roots in , .

I cannot find an example from my lecture notes about primitive roots in . What is the definition of primitive roots in ?

The answers are 40 and none respectively. - Apr 25th 2012, 07:42 AMSylvia104Re: Find the number of primitive roots in Z_101, Z_12
A primitive root in is an integer such that for each such that there exists an integer such that (It follows that

When is a prime, always has primitive roots. Indeed is a cyclic group of order generated by any primitive root; hence the number of primitive roots is This answers your question for

If is not prime, things are a little complicated. For though, you can easily see that for which are the integers coprime with Hence has no primitive roots since there is are no integers such that even though - Apr 26th 2012, 06:49 AMmath2011Re: Find the number of primitive roots in Z_101, Z_12
Thank you. I see. This is really the same as primitive roots in and , just in different notation.