5. In the ring Z[V2] = {a + bv2: a, b e Z} define a norm by N(a + bv2) = |(a + b/2) . (a - by2)| = |a² - 26*|. (a) Prove that this norm is multiplicative. Hint: First show that (a+bv2)(c+dv2): (ac + 2bd) + (ad + be) /2 and (a - by2)(c-dv2) (ac + 2bd) - (ad + bc)V2. (b) Prove N(a + b/2) = 0 if and only if a = b 0. (c) Prove N(a + bv2) = 1 if and only if a + by2 is a unit. %3D %3D %3D
5. In the ring Z[V2] = {a + bv2: a, b e Z} define a norm by N(a + bv2) = |(a + b/2) . (a - by2)| = |a² - 26*|. (a) Prove that this norm is multiplicative. Hint: First show that (a+bv2)(c+dv2): (ac + 2bd) + (ad + be) /2 and (a - by2)(c-dv2) (ac + 2bd) - (ad + bc)V2. (b) Prove N(a + b/2) = 0 if and only if a = b 0. (c) Prove N(a + bv2) = 1 if and only if a + by2 is a unit. %3D %3D %3D
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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a,b,c please
![5. In the ring Z[V2] = {a + bv2:a,b € Z} define a norm by N(a + bv2) = |(a +bv2) ·
(a – by2)| = |a² – 26²|.
(a) Prove that this norm is multiplicative. Hint: First show that (a+bv2)(c+dv2) =
(ac + 2bd) + (ad + be) /2 and (a – by2)(c – dv2) = (ac+ 2bd) – (ad + bc)/2.
(b) Prove N(a + b/2) = 0 if and only if a = b = 0.
(c) Prove N(a + b/2) = 1 if and only if a + bv2 is a unit.
(d) Show that the division algorithm holds in Z[v2]. Hint: Let a, 3 e Z[v2] with
B # 0. Write a/B = r+ sv2, where r, s e Q. Choose m,n e Z such that
|r-m| <1/2 and |s – n| < 1/2. So N(a/B–m-n/2) = |(r – m)² – 2(s – n)²| <
2(1/2)2 = 1/2.
(e) Show that the numbers +(1+ v2)", where n e Z, are units in Z[V2].
%3D](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6d040c73-5bcf-4909-aa2b-abaad050d675%2Fc992d395-1233-4900-aed4-7dc22af72e37%2Fzfyr4yo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:5. In the ring Z[V2] = {a + bv2:a,b € Z} define a norm by N(a + bv2) = |(a +bv2) ·
(a – by2)| = |a² – 26²|.
(a) Prove that this norm is multiplicative. Hint: First show that (a+bv2)(c+dv2) =
(ac + 2bd) + (ad + be) /2 and (a – by2)(c – dv2) = (ac+ 2bd) – (ad + bc)/2.
(b) Prove N(a + b/2) = 0 if and only if a = b = 0.
(c) Prove N(a + b/2) = 1 if and only if a + bv2 is a unit.
(d) Show that the division algorithm holds in Z[v2]. Hint: Let a, 3 e Z[v2] with
B # 0. Write a/B = r+ sv2, where r, s e Q. Choose m,n e Z such that
|r-m| <1/2 and |s – n| < 1/2. So N(a/B–m-n/2) = |(r – m)² – 2(s – n)²| <
2(1/2)2 = 1/2.
(e) Show that the numbers +(1+ v2)", where n e Z, are units in Z[V2].
%3D
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