1. Let d€ 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[√α], 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]. (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 d€ 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[√α], 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]. (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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Can you answer only d and e please?
![1. Let d€ 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[√α],
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].
(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%2F0e782dbb-d945-4a20-ab5f-7eed6cc1b683%2F9mdoc0f_processed.png&w=3840&q=75)
Transcribed Image Text:1. Let d€ 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[√α],
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].
(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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