Are there primitive roots in Z_32?

In summary, the conversation discussed the topic of primitive roots and how to determine if a given number has primitive roots. It was mentioned that a primitive root exists if the number falls into one of the following categories: (a) n=2,4, (b) n=p^k, or (c) n=2p^k. It was also determined that Z_32 does not have primitive roots, despite fulfilling one of the conditions (b), as the multiplicative group \left(\Bbb Z/2^5\Bbb Z\right)^* is not cyclic. Additionally, it was noted that for a number to have primitive roots, it must be an odd prime.
  • #1
Applejacks
33
0
What are the primitive roots of Z_32?

[itex]\varphi[/itex]([itex]\varphi(32)[/itex])=8

However you must first check that there is a primitive root. A PR exists if
(a) n=2,4
(b) n=p^k
(c)n=2p^k

According to the solutions, Z_32 has no primitive roots. Is this correct? 32=2^5 which fulfills one of the conditions (b) so shouldn't it have PRs?
 
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  • #2
Applejacks said:
What are the primitive roots of Z_32?

[itex]\varphi[/itex]([itex]\varphi(32)[/itex])=8

However you must first check that there is a primitive root. A PR exists if
(a) n=2,4
(b) n=p^k
(c)n=2p^k

According to the solutions, Z_32 has no primitive roots. Is this correct? 32=2^5 which fulfills one of the conditions (b) so shouldn't it have PRs?


Well, of course it has no primitive roots as the multiplicative group [itex]\,\left(\Bbb Z/2^5\Bbb Z\right)^*\cong C_2\times C_8\,[/itex] is not cyclic...

DonAntonio
 
  • #3
This is a stupid question but how do I determine it's cyclic? I don't remember covering this in our notes. I was looking at more examples, Z_97,98 and 99 and was wondering what approach I would need to take then.

97=prime=p
98=2 x 7^2=2p^k
99=3^2 x 11=/=2p^k or p^k

97 and 98 have PRs but 99 does not.
 
  • #4
Applejacks said:
This is a stupid question but how do I determine it's cyclic? I don't remember covering this in our notes. I was looking at more examples, Z_97,98 and 99 and was wondering what approach I would need to take then.

97=prime=p
98=2 x 7^2=2p^k
99=3^2 x 11=/=2p^k or p^k

97 and 98 have PRs but 99 does not.


It's not a stupid question at all, and it is not trivial. I could tell you the different cases depending on m in Z_m but you can either

find it in almost any decent group theory book or even google it by "group of units modulo m".

About your results: they are correct as (Z_m)* is cyclic iff [itex]\,m = 2,4,p^k, 2p^k\,[/itex] , with p a prime number and k a natural one.

DonAntonio
 
  • #5
Applejacks said:
What are the primitive roots of Z_32?
A PR exists if
(a) n=2,4
(b) n=p^k
(c)n=2p^k

According to the solutions, Z_32 has no primitive roots. Is this correct? 32=2^5 which fulfills one of the conditions (b) so shouldn't it have PRs?
(b) and (c) are only for odd p.
 
  • #6
haruspex said:
(b) and (c) are only for odd p.

ugh I forgot about that. Every prime except 2 in an odd number so that left my mind. Thanks to both of you
 

1. What are primitive roots of Z_32?

Primitive roots of Z_32 are the numbers that generate the entire group of integers modulo 32 when raised to different powers. These numbers have the property that the order of the group is equal to the order of the primitive root.

2. How many primitive roots are there in Z_32?

There are 8 primitive roots in Z_32, which are 1, 3, 5, 11, 13, 17, 19, and 27. These numbers are relatively prime to 32 and when raised to different powers, they generate all the integers from 1 to 31.

3. What is the order of primitive roots in Z_32?

The order of primitive roots in Z_32 is 16. This means that when a primitive root is raised to the power of 16, it will yield the congruence of 1 modulo 32. Additionally, the order of the group generated by a primitive root is also 16.

4. How can I find the primitive roots of Z_32?

To find the primitive roots of Z_32, you can use the primitive root theorem. This theorem states that for any prime number p, there exists at least one primitive root in Z_p. Therefore, to find the primitive roots of Z_32, you can check the numbers 1 to 31 and see which ones satisfy the primitive root conditions.

5. Why are primitive roots important in Z_32?

Primitive roots are important in Z_32 because they have a special property that makes them useful in various applications, such as cryptography and number theory. They are also important in understanding the structure and properties of finite groups, as they generate the entire group when raised to different powers.

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