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].

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.5: The Binomial Theorem

Problem 50E

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