Let a e Zz be a zero of the irreducible polynomial p(x) = x³ + x² + 2 in Z3[x].a. What is |Z3(a)|? (Z3(a), +) is isomorphic to what abelian group? Give the form of anelement in Z3(a).b. Show that 2a2 +1 is also a zero of p(r).= 3 for all BE Z3(a) \ Z3.c. Explain why deg(B, Z3)d. Determine irr(2a, Z3).

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Answered: Let a € Zz be a zero of the irreducible…

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 7E

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Let a € Zz be a zero of the irreducible polynomial p(x) = x3 + x² + 2 in Z3[x]. a. What is |Z3(a)|? (Z3(a), +) is isomorphic to what abelian group? Give the form of an element in Z3(a). b. Show that 2a2 +1 is also a zero of p(x). c. Explain why deg(8, Z3) = 3 for all BE Z3(a) \ Zg.