Exercise 1. This software lists all idempotents (see the
Chapter 13, Problem 2CE(choose chapter or problem)
Problem 2CE
Exercise 1. This software lists all idempotents (see the chapter exercises for the definition) in Z n . Run the program for various values of n . Use these data to make conjectures about the number of idempotents in Z n as a function of n. For example, how many idempotents are there when n is a prime power? What about when n is divisible by exactly two distinct primes? In the case where n is of the form pq where p and q are primes can you see a relationship between the two idempotents that are not 0 and 1? Can you see a relationship between the number of idempotents for a given n and the number of distinct prime divisors of n ?Exercise 2. This software lists all nilpotent elements (see the chapter exercises for definition) in Z n . Run your program for various values of n . Use these data to make conjectures about nilpotent elements in Z n as a function of n .Exercise 3. This software determines which rings of the form Z p [ i ] are fields. Run the program for all primes up to 37. From these data, make a conjecture about the form of the primes that yield a field.Exercise 1. This software lists all idempotents (see the chapter exercises for the definition) in Z n . Run the program for various values of n . Use these data to make conjectures about the number of idempotents in Z n as a function of n. For example, how many idempotents are there when n is a prime power? What about when n is divisible by exactly two distinct primes? In the case where n is of the form pq where p and q are primes can you see a relationship between the two idempotents that are not 0 and 1? Can you see a relationship between the number of idempotents for a given n and the number of distinct prime divisors of n ?Exercise 2. This software lists all nilpotent elements (see the chapter exercises for definition) in Z n . Run your program for various values of n . Use these data to make conjectures about nilpotent elements in Z n as a function of n .Exercise 3. This software determines which rings of the form Z p [ i ] are fields. Run the program for all primes up to 37. From these data, make a conjecture about the form of the primes that yield a field.
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