In the ring Z[x] prove (a) 2 and 3x + 2 are relatively prime (their only common divisors are units, in this case, ±1) (b) 1 cannot be written as a linear combination of 2 and 3x + 2 in Z[r], that is, there are no integer polynomials p(x) and q(x) such that 1 p(x)2 + q(x) (3x + 2) =
In the ring Z[x] prove (a) 2 and 3x + 2 are relatively prime (their only common divisors are units, in this case, ±1) (b) 1 cannot be written as a linear combination of 2 and 3x + 2 in Z[r], that is, there are no integer polynomials p(x) and q(x) such that 1 p(x)2 + q(x) (3x + 2) =
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 2 college level
![1. In the ring \( \mathbb{Z}[x] \) prove
(a) 2 and \( 3x + 2 \) are relatively prime (their only common divisors are units, in this case, \( \pm 1 \))
(b) \( \pm 1 \) cannot be written as a linear combination of 2 and \( 3x + 2 \) in \( \mathbb{Z}[x] \), that is, there are no integer polynomials \( p(x) \) and \( q(x) \) such that \( \pm 1 = p(x) \cdot 2 + q(x) \cdot (3x + 2) \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32f77ee0-291c-46d0-b315-80fb2fd096d8%2Ffbb00e68-2138-4837-8515-9a5fd59dec59%2Fx0mkmmq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. In the ring \( \mathbb{Z}[x] \) prove
(a) 2 and \( 3x + 2 \) are relatively prime (their only common divisors are units, in this case, \( \pm 1 \))
(b) \( \pm 1 \) cannot be written as a linear combination of 2 and \( 3x + 2 \) in \( \mathbb{Z}[x] \), that is, there are no integer polynomials \( p(x) \) and \( q(x) \) such that \( \pm 1 = p(x) \cdot 2 + q(x) \cdot (3x + 2) \)
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