(d) Show that the division algorithm holds in Z[V2]. Hint: Let a, 3 e Z[v2] with B+ 0. Write a/3 =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].
(d) Show that the division algorithm holds in Z[V2]. Hint: Let a, 3 e Z[v2] with B+ 0. Write a/3 =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].
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.3: The Characteristic Of A Ring
Problem 7E
Related questions
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d,e 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%2F2a54f150-d53a-4b36-89bc-2389355856a4%2Fjzng5nn_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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