Let F be a field and let I = {f(x)∈F[x] | f (a) = 0 for all a in F}.Prove that I is an ideal in F[x]. Prove that I is infinite when F is finiteand I = {0} when F is infinite. When F is finite, find a monicpolynomial g(x) such that I = <g(x)>.

Let F be a field and let I = {f(x)∈F[x] | f (a) = 0 for all a in F}.Prove that I is an ideal in F[x]. Prove that I is infinite when F is finiteand I = {0} when F is infinite. When F is finite, find a monicpolynomial g(x) such that I = <g(x)>.

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

Let F be a field and let I = {f(x)∈F[x] | f (a) = 0 for all a in F}.
Prove that I is an ideal in F[x]. Prove that I is infinite when F is finite
and I = {0} when F is infinite. When F is finite, find a monic
polynomial g(x) such that I = <g(x)>.

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