PMATH 340 Number Theory, Assignment 4 Due: Tues July 21
Read Chapters 6 and 7 in the Lecture Notes, and work on the Exercises for Chapters 6 and 7 in the Practice
Problems. Then solve each of the following problems.
1: (a) Factor 3 + 2
√
3 i as a product of irreducible elements in the ring Z
[√
3 i
]
.
(b) Show that 4 +
√
5 i is irreducible but not prime in the ring Z
[√
5 i
]
.
(c) Show that the ring Z
[√
2
]
is a unique factorization domain.
2: (a) Without proof, list all of the irreducible elements z ∈ Z[√6 i] with ||z|| ≤ 10.
(b) Without proof, list all of the elements z ∈ Z[√6 i] with ||z|| ≤ 10 which do not factor uniquely.
(c) Let p be a prime in Z+. Show that p is reducible in Z[
√
6 i] if and only if p = x2+6y2 for some x, y ∈ Z+.
3: (a) Express the (periodic) continued fraction [1, 3, 1, 1, 2 ] as a quadratic irrational.
(b) Find the 4th convergent c4 =
p4
q4
for the continued fraction representation of e2.
(c) Express
√
43 as a continued fraction and find the smallest unit u > 1 in Z[
√
43 ].