1. Let de Z\{0, 1} be square-free. Recall that Z[√] = {s+t√ds,t = Z} with addition and multiplication inherited from C, is an integral domain (Lemma 7.11 from lectures). Let N: Z[√] \{0} → Z be the norm on Z[√d], given by N(s+t√d) = |s² — dt²|. - (a) Show that N(ab) = N(a)N(b) for all a, b = Z[√] \ {0}. (b) Show that, for a € Z[√d] \ {0}, N(a) = 1 if and only if a is a unit. (c) Find all units of Z[√-5].
1. Let de Z\{0, 1} be square-free. Recall that Z[√] = {s+t√ds,t = Z} with addition and multiplication inherited from C, is an integral domain (Lemma 7.11 from lectures). Let N: Z[√] \{0} → Z be the norm on Z[√d], given by N(s+t√d) = |s² — dt²|. - (a) Show that N(ab) = N(a)N(b) for all a, b = Z[√] \ {0}. (b) Show that, for a € Z[√d] \ {0}, N(a) = 1 if and only if a is a unit. (c) Find all units of Z[√-5].
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.4: Ordered Integral Domains
Problem 8E: If x and y are elements of an ordered integral domain D, prove the following inequalities. a....
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a, b and c please
![1. Let de Z\{0, 1} be square-free. Recall that
Z[√] = {s+t√ds,t = Z}
with addition and multiplication inherited from C, is an integral domain
(Lemma 7.11 from lectures). Let N: Z[√] \{0} → Z be the norm on Z[√d],
given by N(s+t√d) = |s² — dt²|.
-
(a) Show that N(ab) = N(a)N(b) for all a, b = Z[√] \ {0}.
(b) Show that, for a € Z[√d] \ {0}, N(a) = 1 if and only if a is a unit.
(c) Find all units of Z[√-5].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd0a2eb85-71bc-41a8-8f67-a7e3c7209cc3%2Fd0a578aa-b5ef-4541-be0d-ce851f75589d%2Ff0e1puk_processed.png&w=3840&q=75)
Transcribed Image Text:1. Let de Z\{0, 1} be square-free. Recall that
Z[√] = {s+t√ds,t = Z}
with addition and multiplication inherited from C, is an integral domain
(Lemma 7.11 from lectures). Let N: Z[√] \{0} → Z be the norm on Z[√d],
given by N(s+t√d) = |s² — dt²|.
-
(a) Show that N(ab) = N(a)N(b) for all a, b = Z[√] \ {0}.
(b) Show that, for a € Z[√d] \ {0}, N(a) = 1 if and only if a is a unit.
(c) Find all units of Z[√-5].
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