Exercise 1. This software lists
all idempotents (see the chapter exercises for the definition) in
Run the program for various values of n. Use these data to make
conjectures about the number of idempotents in Zn as
n. For example, how many idempotents are there when
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
Run your program for various values of n . Use these data to make
conjectures about nilpotent elements in Zn
as a function of n.
Exercise 3. This software determines which
rings of the form Zp[i] are fields. Run
for all primes up to 37. From these data, make a conjecture about the
form of the primes that yield a field.