I EXAMPLE 4 Let R[x] denote the group of all polynomials with real coefficients under addition. For any f in R[x], let f' denote the derivative of f. Then the mapping f →f' is a homomorphism from R[x] to itself. The kernel of the derivative mapping is the set of all constant polynomials.

I EXAMPLE 4 Let R[x] denote the group of all polynomials with real coefficients under addition. For any f in R[x], let f' denote the derivative of f. Then the mapping f →f' is a homomorphism from R[x] to itself. The kernel of the derivative mapping is the set of all constant polynomials.

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

Prove that the mapping given in Example 4 is a homomorphism.

I EXAMPLE 4 Let R[x] denote the group of all polynomials with real
coefficients under addition. For any f in R[x], let f' denote the derivative
of f. Then the mapping f →f' is a homomorphism from R[x] to itself.
The kernel of the derivative mapping is the set of all constant
polynomials.

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