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 y is a ring homomorphism. (3) Find a polynomial p(x) such that ker o = (p(x)). (4) Find a subring of M2(R) isomorphic to the quotient Q[x]/{p(x)).
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 y is a ring homomorphism. (3) Find a polynomial p(x) such that ker o = (p(x)). (4) Find a subring of M2(R) isomorphic to the quotient Q[x]/{p(x)).
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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![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fab68ef6f-9760-4c1f-88a8-6e5ceb2168b2%2F9ec63e0e-3a42-4381-b457-311ab9a4b892%2F31d2zcc_processed.png&w=3840&q=75)
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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