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Exercise 2.98.1 Let F be any field, and let a E F be arbitrary. Show that the function f: Fx] F that sends r to a and more generally p(x) -> to p(a) is a surjective ring homomorphism whose kernel is the ideal generated by (r- a).

Exercise 2.98.1 Let F be any field, and let a E F be arbitrary. Show that the function f: Fx] F that sends r to a and more generally p(x) -> to p(a) is a surjective ring homomorphism whose kernel is the ideal generated by (r- a).

Advanced Engineering Mathematics
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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Example 2.98 After seeing in Examples 2.96 and 2.97 above how long division can be used to determine kernels of homomorphisms from Qr to other rings, the following should be easy: CHAPTER 2 RINGS AND FIELDS 186 Exercise 2.98.1 Let F be any field, and let a E F be arbitrary. Show that the function f: Fx F that sends x to a and more generally p(x) to p(a) is a surjective ring homomorphism whose kernel is the ideal generated by (x - a).
Transcribed Image Text:Example 2.98 After seeing in Examples 2.96 and 2.97 above how long division can be used to determine kernels of homomorphisms from Qr to other rings, the following should be easy: CHAPTER 2 RINGS AND FIELDS 186 Exercise 2.98.1 Let F be any field, and let a E F be arbitrary. Show that the function f: Fx F that sends x to a and more generally p(x) to p(a) is a surjective ring homomorphism whose kernel is the ideal generated by (x - a).
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