4. Let R = {m + n/2 | m, n e Z}. (a) Show that m + n/2 is a unit in R if and only if m² – 2n2 =±1. Hint: Show that if (m +n/2)(x + y/2) = 1, then (m –n/2)(x – y/2) multiply the two equations. = 1 and - |
4. Let R = {m + n/2 | m, n e Z}. (a) Show that m + n/2 is a unit in R if and only if m² – 2n2 =±1. Hint: Show that if (m +n/2)(x + y/2) = 1, then (m –n/2)(x – y/2) multiply the two equations. = 1 and - |
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please answer part a.
Transcribed Image Text:4. Let R = {m + n/2 | m, n e Z}.
(a) Show that m +n/2 is a unit in R if and only if m2 – 2n? =±1.
Hint: Show that if (m +n/2)(x + y/2) = 1, then (m – n/2)(x – yv2)
multiply the two equations.
= 1 and
(b) Show that 1+ 2/2 has infinite order in R*.
(c) Show that 1 and –1 are the only units that have finite order in R×.
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