Exercise 2.114 Consider the ring Q[/m] of Example 2.12. Now assume for this exercise that m is not a perfect square. Show that a + b/m = 0 (for a and b in %3D Q) if and only if a = b= 0. Show that Q[/m] is a field.
Exercise 2.114 Consider the ring Q[/m] of Example 2.12. Now assume for this exercise that m is not a perfect square. Show that a + b/m = 0 (for a and b in %3D Q) if and only if a = b= 0. Show that Q[/m] is a field.
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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abstract algebra
![**Exercise 2.114**
Consider the ring \( \mathbb{Q}[\sqrt{m}] \) of Example 2.12. Now assume for this exercise that \( m \) is not a perfect square. Show that \( a + b\sqrt{m} = 0 \) (for \( a \) and \( b \) in \( \mathbb{Q} \)) if and only if \( a = b = 0 \). Show that \( \mathbb{Q}[\sqrt{m}] \) is a field.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d940ce8-cba2-4a95-af25-aae0739ca5aa%2F4667082b-eaf6-45fc-91e0-1fd4f4bd89a5%2Fbr6jmfj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise 2.114**
Consider the ring \( \mathbb{Q}[\sqrt{m}] \) of Example 2.12. Now assume for this exercise that \( m \) is not a perfect square. Show that \( a + b\sqrt{m} = 0 \) (for \( a \) and \( b \) in \( \mathbb{Q} \)) if and only if \( a = b = 0 \). Show that \( \mathbb{Q}[\sqrt{m}] \) is a field.
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