MAT 563 HOMEWORK 8: Due Friday, March 26
(1) Suppose R is a PID, and I ⊂ R is a nonzero ideal.
(a) Prove that I is a prime ideal if and only if I is generated by a
prime element of R.
(b) Show that if I is a nonzero prime ideal, then it is a maximal
ideal, so that R/I is a field.
(2) Let F be a field and let F [[X]] be the ring of formal power series in
X. An element of this ring has the form
∞
X
ai X i = a0 + a1 X + a2 X 2 + · · · ,
i=0
where ai ∈ F . (There are no convergence considerations here.)
(a) Show that F [[X]]∗ is the set of power series with nonzero constant term.
(b) Show that F [[X]] is a PID.
(3) Let D be a square-free
integer (i.e. a2 |D ⇐⇒ a2 = 1).
√
(a) For x ∈ Z[ D], suppose √N (x) = p is a prime integer. Prove
that x is irreducible
√ in Z[ D].
√
√
(b) Show that 3, 2+ −5, and 2− √
−5 are all irreducible in Z[ −5].
Explain how this shows that Z[ −5] is a domain which is not a
UFD.
Remark: A famous conjecture of Gauss
√ (finally proven in 1952/1969
by Heegner/Stark) asserts that Z[ −D] is a PID only for
D = 1, 2, 3, 7, 11, 19, 43, 67, 163.
(4) Let R be an integral domain. Given a, b ∈ R, a greatest common
divisor of a and b is an element d ∈ R satisfying:
• d|a and d|b
• If x|a and x|b, then x|d.
(a) Suppose d is a gcd of a and b. Prove that any other gcd is
associate to d, i.e. of the form ud, where u is a unit.
(b) Suppose R is a UFD. Prove that any two elements have a gcd.
(c) Two elements a, b ∈ R are relatively prime if 1 is a gcd of a
and b. Suppose R is a PID. Prove that a and b are relatively
prime if and only if
1 = ar + bs
for some r, s ∈ R.
Bonus: Show that the above may not be true in a UFD. (Use an
example from class.)
√
(5) Consider the ring of Gaussian integers R = Z[i], where i = −1. You
may take as given the fact that this ring is a PID (see pages 389-390).
Also since R is a domain, recall that an element is irreducible if and
only if it is prime.
(a) Find all units of Z[i]. (Use a previous HW problem.)
(b) Prove that 2 is not prime in Z[i]. Factor it into irreducibles in
Z[i].
(c) Let p ∈ Z be an odd prime number (here and below). Note that
p may or may not be prime in Z[i]. Prove that p is prime in Z[i]
if and only if p cannot be expressed as a sum of two squares in
Z. (For example, 3 is prime in Z[i], but 5 = 12 + 22 is not.)
(d) Prove that if p is a sum of two squares, then p ≡ 1 mod 4.
(e) Prove that if p ≡ 1 mod 4, then p|(n2 + 1) for some integer n.
(Hint: Use the fact (to be proven next week) that (Z/pZ)∗ is a
cyclic group of order p − 1. Explain why there is an element n
of order 4 in this group.)
(f) Prove that if p|(n2 + 1) for some integer n, then p is not prime
in Z[i].
You have proven the following classical theorem from number
theory: For an odd prime p > 0, the following are equivalent:
(i) p is a sum of two squares
(ii) p ≡ 1 mod 4
(iii) p is not prime in Z[i].
(iv) p divides an integer of the form n2 + 1.