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Answer: Exercise 1. This software lists all idempotents
Chapter 13, Problem 3CE(choose chapter or problem)
Problem 3CE
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.
Questions & Answers
QUESTION:
Problem 3CE
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.ANSWER:
Step 1 of 2
A ring R is a set with two binary operations, addition and multiplication, satisfying several properties: R is an Abelian group under addition, and the multiplication operation satisfies the associative law
And distributive laws
And
For every
The identity of the addition operation is denoted 0. If the multiplication operation has an identity, it is called a unity. If multiplication is commutative, we say that R is commutative. Let a be an element of a ring with unity. If a has a multiplicative inverse, we say that a is a unit, and denote the multiplicative inverse
If R is a commutative ring with unity and every nonzero element of R is a unit, we say that R is a field.
The ring 
Under addition and multiplication modulo n. It is a commutative ring with 1 as the unity.
Addition and multiplication are defined as they are for complex numbers except that all arithmetic is modulo n.