Formula for the Number of Units in a Ring Modulo a Prime Power

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In summary, the conversation is discussing finding a formula for the number of units in the ring Z/p^nZ, denoted by |(Z/p^nZ)^x|. The use of "|" symbols around the notation indicates the number of elements in the set. The question is asking for clarification on the notation and if it refers to the absolute value or elements in the set.
  • #1
mathmajor2013
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So the question is:

Suppose p is a prime and n is a positive integer. Find a formula for |(Z/p^nZ)^x|.

I do not know what this notation means, what do the | | mean around this? I know that the other part is the set of all the units in Z/p^nZ, but I have no idea what the | | mean. I don't think it's absolute value. Is it just the elements of it? Thanks for the help.
 
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  • #2
mathmajor2013 said:
I do not know what this notation means, what do the | | mean around this?

"The number of elements in". |{2, 4, 6}| = 3.
 
  • #3
Do you mean this:

[tex]|({\textbf{Z}} / p^{n}{\textbf{Z}})^{*}| [/tex]

in words: the number of units in the ring [tex]{\textbf{Z}} / p^{n}{\textbf{Z}}[/tex]

(suggestions: if would like to avoid the tex tag you can use the sup tag to write down your statements: |(Z/pnZ)x|
)
 
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1. What is the "FORMULA FOR |(Z/p^nZ)^x|"?

The formula for |(Z/p^nZ)^x| is used in number theory and abstract algebra to calculate the order of elements in a quotient group. It is defined as |(Z/p^nZ)^x| = p^(n-x), where p is a prime number, n is a positive integer, and x is an element in the quotient group.

2. How is the formula derived?

The formula is derived from the cyclic structure of the quotient group (Z/p^nZ)^x, which is a group of integers modulo p^n. By applying basic principles of group theory, it can be shown that the order of elements in this group is related to the power of the prime number p raised to the difference between the quotient group's dimension and the element's order.

3. What is the significance of the formula?

The formula is significant because it allows us to quickly and easily calculate the order of elements in a quotient group, which is a fundamental property of any group. This can be useful in various mathematical applications, such as cryptography and number theory.

4. Can the formula be applied to other types of groups?

Yes, the formula can be applied to other types of groups, as long as they have a similar cyclic structure and follow the same principles of group theory. However, the specific values of p, n, and x may differ depending on the group's structure.

5. Are there any limitations to using the formula?

The formula is only applicable to quotient groups with a cyclic structure, and it may not work for more complex groups. Additionally, the values of p, n, and x must be known in order to use the formula, which may not always be the case in practical applications.

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