MATH 323: Rings and Modules
The University of British Columbia
Winter 2015 Term 2
Miljan Brakočević
Homework Assignment 2
This problem set is due on Thursday, January 21, 2016 at the end of class.
Please staple your assignment. Late homework will not be accepted under any circumstances. E-mail submission will not be accepted.
Reading Part (no submission required):
Please read the following in Dummit & Foote (3rd ed):
√
Quadratic Fields Q( D): page 227, item (5)
Quadratic Integer Rings: pages 229, 230
Do the Dummit & Foote (3rd ed) exercises:
Section
Section
Section
Section
7.1:
7.2:
7.3:
7.5:
Exercise 28
Exercises 2, 3, 4, 5(a)
Exercise 10
Exercise 5
Do the following additional ones:
A1. Describe the center of the ring of real Hamilton Quaternions HR .
A2. Let HR be the ring of real Hamilton Quaternions and define N : HR → R≥0 by
N (a + bi + cj + dk) = a2 + b2 + c2 + d2 .
The map N is called a norm. Prove the following:
(a) N (α) = αᾱ for all α ∈ HR , where if α = a + bi + cj + dk then ᾱ = a − bi − cj − dk.
(b) N (αβ) = N (α)N (β) for all α, β ∈ HR .
(c) The inverse of every non-zero element α ∈ HR is
ᾱ
.
N (α)
(d) Consider a subring of HR given by
HZ = {a + bi + cj + dk : a, b, c, d ∈ Z}
and called Integral Hamilton Quaternions. Prove that an element α ∈ HZ is a
unit in HZ if and only if N (α) = 1.
1