1. Let dЄ Z\{0,1} be square-free. Recall that Z[√] = {s+tv +t√d|s, 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[√d] \ {0}. (b) Show that, for a Є Z[√] \ {0}, N(a) = 1 if and only if a is a unit. (c) Find all units of Z[√−5]. (d) Show that 2,1+ √√-5 and 1 - ✓-5 are irreducible elements of Z[√−5]. Hint: Show first that N(y) 2 and N(y) ± 3 for all y € Z[√−5]\{0}. Then apply N to possible factorisations of 2, 1+ √-5 and 1 – √−5 in Z[√−5]. (e) By considering the factorization 6 = (1+√-5)(1-5), show that 2 is not a prime element in Z[5].
1. Let dЄ Z\{0,1} be square-free. Recall that Z[√] = {s+tv +t√d|s, 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[√d] \ {0}. (b) Show that, for a Є Z[√] \ {0}, N(a) = 1 if and only if a is a unit. (c) Find all units of Z[√−5]. (d) Show that 2,1+ √√-5 and 1 - ✓-5 are irreducible elements of Z[√−5]. Hint: Show first that N(y) 2 and N(y) ± 3 for all y € Z[√−5]\{0}. Then apply N to possible factorisations of 2, 1+ √-5 and 1 – √−5 in Z[√−5]. (e) By considering the factorization 6 = (1+√-5)(1-5), show that 2 is not a prime element in Z[5].
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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Question
![1. Let dЄ Z\{0,1} be square-free. Recall that
Z[√] = {s+tv
+t√d|s, 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[√d] \ {0}.
(b) Show that, for a Є Z[√] \ {0}, N(a) = 1 if and only if a is a unit.
(c) Find all units of Z[√−5].
(d) Show that 2,1+ √√-5 and 1 - ✓-5 are irreducible elements of Z[√−5].
Hint: Show first that N(y) 2 and N(y) ± 3 for all y € Z[√−5]\{0}. Then
apply N to possible factorisations of 2, 1+ √-5 and 1 – √−5 in Z[√−5].
(e) By considering the factorization 6 = (1+√-5)(1-5), show that 2 is not
a prime element in Z[5].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd0a2eb85-71bc-41a8-8f67-a7e3c7209cc3%2Fffd69b78-0be8-450f-a9d2-f1b57405daf9%2Fl7qv71v_processed.png&w=3840&q=75)
Transcribed Image Text:1. Let dЄ Z\{0,1} be square-free. Recall that
Z[√] = {s+tv
+t√d|s, 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[√d] \ {0}.
(b) Show that, for a Є Z[√] \ {0}, N(a) = 1 if and only if a is a unit.
(c) Find all units of Z[√−5].
(d) Show that 2,1+ √√-5 and 1 - ✓-5 are irreducible elements of Z[√−5].
Hint: Show first that N(y) 2 and N(y) ± 3 for all y € Z[√−5]\{0}. Then
apply N to possible factorisations of 2, 1+ √-5 and 1 – √−5 in Z[√−5].
(e) By considering the factorization 6 = (1+√-5)(1-5), show that 2 is not
a prime element in Z[5].
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