Find Number of Generators in Z_(p^r)

  • Thread starter Eivind
  • Start date
  • Tags
    Generators
In summary, Z_(p^r) is a set of integers modulo p^r, where p is a prime number and r is a positive integer. The number of generators in this set is equal to the number of positive integers less than p^r that are relatively prime to p^r, and can be calculated using Euler's totient function. This information is important for various mathematical applications and can change depending on the values of p and r. While there is no general formula for calculating the number of generators, it can be determined using mathematical techniques such as Euler's totient function or the Chinese remainder theorem.
  • #1
Eivind
29
0
Let p be a prime number. Find the number of generators of the cyclic group Z_(p^r), where r is an integer >=1.

A small hint, please?
 
Physics news on Phys.org
  • #2
What do you know about generators of cyclic groups?
 
  • #3
Well, the generators are those which are relatively prime to p^r.
 
  • #4
Eivind said:
Well, the generators are those which are relatively prime to p^r.
And don't you know an algorithm for computing how many integers in the range [0, n] are relatively prime to any number n?
 

What is Z_(p^r)?

Z_(p^r) is a mathematical notation that represents the set of integers modulo p^r, where p is a prime number and r is a positive integer. It includes all the numbers from 0 to p^r-1, and follows specific rules for addition and multiplication.

How do you find the number of generators in Z_(p^r)?

The number of generators in Z_(p^r) is equal to the number of positive integers less than p^r that are relatively prime to p^r. This can be calculated using Euler's totient function, which counts the number of positive integers less than a given number that are coprime to it.

Why is it important to find the number of generators in Z_(p^r)?

Knowing the number of generators in Z_(p^r) is important in various mathematical applications, such as cryptography and number theory. It can also help us understand the structure and properties of this set, and can be used in solving related problems.

Can the number of generators in Z_(p^r) change?

Yes, the number of generators in Z_(p^r) can change depending on the values of p and r. For example, if p is a small prime number, there may be many generators in Z_(p^r), but as p increases or r becomes larger, the number of generators may decrease.

Is there a formula for calculating the number of generators in Z_(p^r)?

There is no general formula for calculating the number of generators in Z_(p^r), but it can be determined by using various mathematical techniques such as Euler's totient function or the Chinese remainder theorem. The exact method used may depend on the specific values of p and r.

Similar threads

  • Linear and Abstract Algebra
Replies
17
Views
4K
  • Linear and Abstract Algebra
Replies
33
Views
3K
  • Linear and Abstract Algebra
Replies
11
Views
1K
  • Linear and Abstract Algebra
Replies
7
Views
2K
  • Linear and Abstract Algebra
Replies
5
Views
1K
  • Linear and Abstract Algebra
Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
719
  • Linear and Abstract Algebra
Replies
1
Views
693
  • Linear and Abstract Algebra
Replies
1
Views
616
  • Linear and Abstract Algebra
Replies
2
Views
1K
Back
Top