LetΦ be the ring homomorphism from Z[x] to Z given by Φ(f(x)) =f(1). Find a polynomial g(x) in Z[x] such that Ker Φ =<g(x)>. Isthere more than one possibility for g(x)? To what familiar ring isZ[x]/Ker Φ isomorphic? Do this exercise with Z replaced by Q.

LetΦ be the ring homomorphism from Z[x] to Z given by Φ(f(x)) =f(1). Find a polynomial g(x) in Z[x] such that Ker Φ =<g(x)>. Isthere more than one possibility for g(x)? To what familiar ring isZ[x]/Ker Φ isomorphic? Do this exercise with Z replaced by Q.

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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Question

LetΦ be the ring homomorphism from Z[x] to Z given by Φ(f(x)) =
f(1). Find a polynomial g(x) in Z[x] such that Ker Φ =<g(x)>. Is
there more than one possibility for g(x)? To what familiar ring is
Z[x]/Ker Φ isomorphic? Do this exercise with Z replaced by Q.

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