Exercise 2.106.2 Let F be a field, and let a be a nonzero element of F. Let b be an arbitrary element of F. Prove that the map f: Fa]→ Fr] that -> sends x to ax +b and more generally, a polynomial po+P1x++ Pn to the polynomial po + P1(ax +b) + .+ pn(ax + b)" is an automorphism of F[a].
Exercise 2.106.2 Let F be a field, and let a be a nonzero element of F. Let b be an arbitrary element of F. Prove that the map f: Fa]→ Fr] that -> sends x to ax +b and more generally, a polynomial po+P1x++ Pn to the polynomial po + P1(ax +b) + .+ pn(ax + b)" is an automorphism of F[a].
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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absract algebra
![**Exercise 2.106.2**
Let \( F \) be a field, and let \( a \) be a nonzero element of \( F \). Let \( b \) be an arbitrary element of \( F \). Prove that the map \( f : F[x] \rightarrow F[x] \) that sends \( x \) to \( ax + b \) and more generally, a polynomial \( p_0 + p_1x + \cdots + p_nx^n \) to the polynomial \( p_0 + p_1(ax + b) + \cdots + p_n(ax + b)^n \) is an automorphism of \( F[x] \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d940ce8-cba2-4a95-af25-aae0739ca5aa%2F5e93c442-5914-4872-8494-123691c7b43a%2Fgbbftm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise 2.106.2**
Let \( F \) be a field, and let \( a \) be a nonzero element of \( F \). Let \( b \) be an arbitrary element of \( F \). Prove that the map \( f : F[x] \rightarrow F[x] \) that sends \( x \) to \( ax + b \) and more generally, a polynomial \( p_0 + p_1x + \cdots + p_nx^n \) to the polynomial \( p_0 + p_1(ax + b) + \cdots + p_n(ax + b)^n \) is an automorphism of \( F[x] \).
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