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# Exercise 1. This software lists all idempotents (see the ISBN: 9781133599708 52

## Solution for problem 2CE Chapter 13

Contemporary Abstract Algebra | 8th Edition

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Problem 2CE

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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##### ISBN: 9781133599708

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Exercise 1. This software lists all idempotents (see the