advertisement

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