1. In this exercise we will prove that Z[√-5] is an integral domain but not a unique factorisation domain (UFD). Note that in S we have the following factorisations of the element 6: 6=2.3=(1+√√−5) · (1−√−5). It turns out that these are factorisations into irreducible elements. To see this, prove that the element 1+ √-5 is irreducible, by assuming that 1+√√-5= (a+b√√-5). (c+d√√−5) and taking modulus squared. Explain briefly why a similar argument shows that the other three ele- ments 1-5, 2, and 3 are also irreducible.
1. In this exercise we will prove that Z[√-5] is an integral domain but not a unique factorisation domain (UFD). Note that in S we have the following factorisations of the element 6: 6=2.3=(1+√√−5) · (1−√−5). It turns out that these are factorisations into irreducible elements. To see this, prove that the element 1+ √-5 is irreducible, by assuming that 1+√√-5= (a+b√√-5). (c+d√√−5) and taking modulus squared. Explain briefly why a similar argument shows that the other three ele- ments 1-5, 2, and 3 are also irreducible.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.4: Zeros Of A Polynomial
Problem 31E
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 2 steps with 1 images
Recommended textbooks for you
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Algebra for College Students
Algebra
ISBN:
9781285195780
Author:
Jerome E. Kaufmann, Karen L. Schwitters
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,
Algebra for College Students
Algebra
ISBN:
9781285195780
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage