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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