Solving Zn[X] Primality Problem with a Unit in Zn

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In summary, the conversation is about solving a problem involving a unit in Zn, and showing that (X+a)n - (Xn + a) = 0 in Zn[X] if and only if n is prime. The person asking for help is unsure how to use the hint given and how to prove that n is prime. They have attempted to expand the expression but have not found anything useful. They are seeking further assistance.
  • #1
Mathsgirl
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Homework Statement



Hi,

I can't figure out this problem:
Let a in Zn be a unit. Show that (X+a)n - (Xn + a) = 0 in Zn[X] if and only if n is prime.

Thanks for any help :)

Homework Equations



Binomial expansion

The Attempt at a Solution



There is a hint which says to show that if n is not prime then n doesn't divide nCp if p is a prime divisor of n. I'm not sure how to show this or how to use it in the main proof.

If I assume the LHS is true I can expand it using the binomial expansion but I don't seem to get anything useful. I can factorise out an a.
 
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  • #2
I'm not sure how to show that n is prime though. Any help would be much appreciated. Thanks in advance!
 

What is the "Zn[X] Primality Problem"?

The Zn[X] Primality Problem is a mathematical problem that involves determining whether a given number, X, is a prime number in a specific mathematical structure known as Zn. Zn refers to the set of integers modulo n, where n is any positive integer.

What is the significance of solving the Zn[X] Primality Problem?

Solving the Zn[X] Primality Problem has practical applications in cryptography and number theory. It can also help in understanding the structure and properties of prime numbers in different mathematical structures.

What is a "unit" in Zn?

In Zn, a unit is an element that has a multiplicative inverse, meaning it can be multiplied by another element to give the identity element, 1. In other words, for any element a in Zn, there exists an element b such that a*b = 1.

How can a unit in Zn help in solving the Primality Problem?

A unit in Zn can be used to simplify the process of determining whether a given number, X, is a prime in Zn. By finding a unit u in Zn, we can reduce the problem to determining whether X*u is a prime in Zn. This is because if X*u is a prime, then X must also be a prime.

What is the general approach to solving the Zn[X] Primality Problem with a unit?

The general approach involves finding a unit u in Zn and then using it to reduce the Primality Problem to a simpler problem, such as determining the primality of X*u. This can be done using various mathematical techniques and algorithms, such as the Miller-Rabin primality test or the Lucas probable prime test.

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