4. An integer dЄ Z\ {0} is said to be square-free if p² | d for any prime p. Let de Z\{0, 1} be square-free. Show that √√d & Q, where √d=i√√d if d <0. You may assume that if a, b and r are integers, with a, r not both zero, rlab and the greatest common divisor of a and r equal to 1, then r|b. Hint: The difficult case is d > 1. Write √√d = a/b € Q and aim for a contradiction by considering prime factors of a.
4. An integer dЄ Z\ {0} is said to be square-free if p² | d for any prime p. Let de Z\{0, 1} be square-free. Show that √√d & Q, where √d=i√√d if d <0. You may assume that if a, b and r are integers, with a, r not both zero, rlab and the greatest common divisor of a and r equal to 1, then r|b. Hint: The difficult case is d > 1. Write √√d = a/b € Q and aim for a contradiction by considering prime factors of a.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.1: Real Numbers
Problem 35E
Related questions
Question
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 3 steps with 1 images
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra: Structure And Method, Book 1
Algebra
ISBN:
9780395977224
Author:
Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:
McDougal Littell
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra: Structure And Method, Book 1
Algebra
ISBN:
9780395977224
Author:
Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:
McDougal Littell