#
Chapter 13

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*Exercise 1.* This software lists
all idempotents (see the 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*?

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*Exercise 2. *This software lists
all nilpotent elements (see the exercises for the 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.