**Exercise 2. ** Let *Z*_{n}[*i*] = {*
a+bi* | *a, b*
belong to *Z*_{n}, *i*^{2}=-1 }
(the Gaussian integers modulo *n* ). This software finds the group of
units
of this ring and the order of each element of the group. Run the program
for
*n* = 3, 7, 11, and 23. Is the group of units cyclic for these cases?
Try to guess a
formula for the order of the group of units of
*Z*_{n}[*i*] as
a function of *n* when * n*
is a prime and *n* mod 4 = 3.
Run the program for *n* = 9 and 27. Are the groups
cyclic? Try to guess a formula for the order when *n =
3*^{k}.
Run the program for *n* = 5, 13, 17, and 29. Is the group
cyclic for these cases? What is the largest order of any element in
the group?
Try to guess a formula for the order of the
group of units of
*Z*_{n}[*i*] as a function of *
n*
when *n* is a prime and *n* mod 4 = 1.
Try to guess a formula for the largest order of any element in the
group of units of
*Z*_{n}[*i*] as a function of *
n*
when *n* is a prime and *n* mod 4 = 1.
On the basis of the orders of the elements of the group of units, try to
guess the isomorphism class of the group.
Run the program for
*n* = 25. Is this group
cyclic? Based on the number of elements in this group and the orders
of the elements, try to guess the isomorphism class of the group.