2.Let IC Q[x] denote the principal ideal generated by x4 – 26x + 52.a. Prove that E :=Q[x]/I is a field.b. Give an example of two cosets p(x)+ I and q(x)+I with deg p > 2 and deg q > 2 suchthat (p(x) + I)(q(x)+I) =1+I.c. Give a basis for E over Q.

Given I⊂ℚx be the principle ideal generated by x4-26x+52. (a). The given polynomial is fx=x4-26x+52.…

Answered: Let IC Q[x] denote the principal ideal…

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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Let IC Q[x] denote the principal ideal generated by x – 26x + 52. Prove that E := Q[x]/I is a field. Give an example of two cosets p(x)+I and q(x)+I with deg p > 2 and deg q > 2 such that (p(x) + I)(q(x)+I) = 1+I. Give a basis for E over Q.