a. Show that x3 + x2 + 1 is irreducible over ℤ3.b. Let ꭤ be a zero of x3 + x2 + 1 in an extension field of ℤ2. Show that x3 + x2 + 1 factors into three linear factors in (ℤ2(ꭤ))[x] by actually finding this factorization.[Hint: Every element of ℤ2(ꭤ) is of the formꭤ0 + ꭤ1ꭤ + ꭤ2ꭤ2 for ꭤi = 0,1.Divide x3 + x2 + 1 by x - ꭤ by long division. Show that the quotient also has a zero in ℤ2(ꭤ) by simply trying the eight possible elements. Then complete the factorization)

(a) The objective is to show that is irreducible over . Let . If , then, If , then, So, has no…

Answered: a. Show that x3 + x2 + 1 is irreducible…

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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a. Show that x³ + x² + 1 is irreducible over ℤ₃.

b. Let ꭤ be a zero of x³ + x² + 1 in an extension field of ℤ₂. Show that x³ + x² + 1 factors into three linear factors in (ℤ₂(ꭤ))[x] by actually finding this factorization.

[Hint: Every element of ℤ₂(ꭤ) is of the form

ꭤ₀ + ꭤ₁ꭤ + ꭤ₂ꭤ² for ꭤ_i = 0,1.

Divide x³ + x² + 1 by x - ꭤ by long division. Show that the quotient also has a zero in ℤ₂(ꭤ) by simply trying the eight possible elements. Then complete the factorization)

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