Answered: Let f(x) = x³ + 6 E Z,[x]. Write f(x)…

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Let f(x) = x³ + 6 E Z,[x]. Write f(x) as a product of irreducible polynomials over Z,.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.4: Zeros Of A Polynomial

Problem 30E

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Question

Let f(x) = x³ + 6 E Z,[x]. Write f(x) as a product of irreducible
polynomials over Z,.

Expert Solution

Step 1

Consider the given polynomial,

Factor theorem for field:

Let F be a field be an element, a polynomial. If f(a) = 0, then (x - a) is factor of f(x).

Find zeros of f(x) over the field Z₇ = {1, 2, 3, 4, 5, 6 }.

Step by step

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