4. The following polynomial is irreducible in Q[x]: m(x) = x² - 2x²-1 € Q[x] Let's consider the ideal I = (m(x)), and the quotient ring F = = Q[x]/I. (a) Explain why F is a field. (b) Consider the element a = x + I of F. Show that a is a root of m(x). In other words, show that is zero in the ring F. a4-2a2-1=0

4. The following polynomial is irreducible in Q[x]: m(x) = x² - 2x²-1 € Q[x] Let's consider the ideal I = (m(x)), and the quotient ring F = = Q[x]/I. (a) Explain why F is a field. (b) Consider the element a = x + I of F. Show that a is a root of m(x). In other words, show that is zero in the ring F. a4-2a2-1=0

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.6: Algebraic Extensions Of A Field

Problem 13E

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