Consider the set S = Z[√2] as defined in Exercise 3.6.6, a set of real numbers. a) Recall that a unit is an element with norm ±1 - what are the units in S? (Note: the norm of an integer x = a +b√r, n(x) is a² - r * b²; so in S, we want all x € S s.t n(x) = 1=a² -2 * 6²) b) Prove that 7 is not prime in S. c) (harder) Prove that 5 is prime in S.
Consider the set S = Z[√2] as defined in Exercise 3.6.6, a set of real numbers. a) Recall that a unit is an element with norm ±1 - what are the units in S? (Note: the norm of an integer x = a +b√r, n(x) is a² - r * b²; so in S, we want all x € S s.t n(x) = 1=a² -2 * 6²) b) Prove that 7 is not prime in S. c) (harder) Prove that 5 is prime in S.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 90E
Related questions
Question
In the second image, is the work i have so far,
![Consider the set S = Z[√2] as defined in Exercise 3.6.6, a set of real numbers.
a) Recall that a unit is an element with norm ±1 - what are the units in S? (Note: the
norm of an integer x = a +b√r, n(x) is a² - r * b²; so in S, we want all x € S s.t
n(x) = 1=a² -2 * 6²)
b) Prove that 7 is not prime in S.
c) (harder) Prove that 5 is prime in S.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb9b6f3f6-7779-4b86-9750-acc8d4a11091%2F05b9839a-af35-4c47-8c3a-6ebe20498c18%2Fdg2ey7j_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the set S = Z[√2] as defined in Exercise 3.6.6, a set of real numbers.
a) Recall that a unit is an element with norm ±1 - what are the units in S? (Note: the
norm of an integer x = a +b√r, n(x) is a² - r * b²; so in S, we want all x € S s.t
n(x) = 1=a² -2 * 6²)
b) Prove that 7 is not prime in S.
c) (harder) Prove that 5 is prime in S.
Transcribed Image Text:a² - 2b² = 1 → a= ± 1 (if b =0) The units of S are 1, and -1
Since, a +b√2 € Z(√2) will be unit of Z(√2) if and only if there exist
some c + d√√2 € Z(√2) such that:
xy = 1 → (a + b√2)(c + d√√2) = 1
ac + ad√2 + bc√2 + 2bd = 1
ac + 2bd + ad√2 + bc√2
(ac +2bd) + √2(ad + bc) = 1
= 1
=
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 7 steps with 6 images
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,