Math 417 Homework 10: Due Friday April 25
Instructions: You can discuss these problems with others,
but write up your solutions on your own (i.e., don't just
copy someone else's solutions, else the feedback I give
you won't help you much). Please be neat and write in full sentences.
Do four of the five problems. Justify your answers.
-  Let 1 < a < b be relatively prime integers. If gcd(phi(a), phi(b)) > 1, show that U(ab) is not cyclic.
[Hint: Apply Cauchy's Theorem.]
-  Let n > 1 be an integer.
- (a) Show that Aut(Zn)
is not cyclic if n is divisible by two different odd primes.
[Hint: Apply problem 1.]
- (b) Show that Aut(Zn)
is not cyclic if n is divisible by both an odd prime and by 4.
-  If n = 2r for an integer r > 0, show that
Aut(Zn) is cyclic if and only if r is 1 or 2.
-  Show that if Aut(Zn)
is cyclic, then one of the following is true:
- (a) n is 1, 2 or 4, or
- (b) n is ps or 2ps for an odd
prime p and a positive integer s.
Aside: It is known that Aut(Zn) is cyclic
in each of the cases
listed in 4(a) and 4(b), so this gives a complete determination
of those n for which Aut(Zn) is cyclic.
[Aside: It is easy to solve the quadratic equation
x2 mod p = 1. There are two solutions in Zp,
namely x = 0 and x = p-1.
The next problem concerns solving x2 mod p = p - 1 in
To do it, you may assume that Aut(Zp) is cyclic
whenever p is a prime.]
-  Let p be a prime.
- (a) Show that x2 mod p = p - 1
has a solution if and only if either p = 2 or p - 1 is divisible by 4.
[Hint: Think of x as being in U(p); then rephrase the problem
in terms of |x|.]
- (b) Find all solutions x to x2 mod p = p - 1
when p is 5, 13, 17 and 29.