Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
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Question
3
Let p: Q[x] → M2(R) be defined by
p(ao + a1 + a2x² + ...+ anx")
= ( )
ao
(1) Write down o(2 +x – x³), y(2a² + 7æ³), p(x² – x).
(2) Show that p is a ring homomorphism.
(3) Find a polynomial p(x) such that ker y = (p(x)).
(4) Find a subring of M2(R) isomorphic to the quotient Q[x]/(p(x)).
(5) Is ker y a prime ideal? A maximal ideal? Justify your answers.
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Transcribed Image Text:Let p: Q[x] → M2(R) be defined by p(ao + a1 + a2x² + ...+ anx") = ( ) ao (1) Write down o(2 +x – x³), y(2a² + 7æ³), p(x² – x). (2) Show that p is a ring homomorphism. (3) Find a polynomial p(x) such that ker y = (p(x)). (4) Find a subring of M2(R) isomorphic to the quotient Q[x]/(p(x)). (5) Is ker y a prime ideal? A maximal ideal? Justify your answers.
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