25. a. Show that x³ + x² + 1 is irreducible over Z₂. b. Let a 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(a))[x] by actually finding this factorization. [Hint: Every element of Z₂(a) is of the form a₁ + a₁ + a₂a² for a₁ = 0, 1. Divide x³ + x² + 1 by x − a by long division. Show that the quotient also has a zero in Z₂(a) by simply trying the eight possible elements. Then complete the factorization.]
25. a. Show that x³ + x² + 1 is irreducible over Z₂. b. Let a 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(a))[x] by actually finding this factorization. [Hint: Every element of Z₂(a) is of the form a₁ + a₁ + a₂a² for a₁ = 0, 1. Divide x³ + x² + 1 by x − a by long division. Show that the quotient also has a zero in Z₂(a) by simply trying the eight possible elements. Then complete the factorization.]
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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![3
25. a. Show that x³ + x² + 1 is irreducible over Z2.
b. Let a 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(a))[x] by actually finding this factorization. [Hint: Every element of Z₂(a) is of the form
a₁ + a₁α + a₂α² for a₁ = 0, 1.
Divide x³ + x² + 1 by x - a by long division. Show that the quotient also has a zero in Z₂(a) by simply
trying the eight possible elements. Then complete the factorization.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8f0df94a-1d02-48d9-8ad0-d3fdfd9735f8%2F1377e9b3-246f-4fc1-8682-a5672a4320b9%2Fitb7tk_processed.png&w=3840&q=75)
Transcribed Image Text:3
25. a. Show that x³ + x² + 1 is irreducible over Z2.
b. Let a 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(a))[x] by actually finding this factorization. [Hint: Every element of Z₂(a) is of the form
a₁ + a₁α + a₂α² for a₁ = 0, 1.
Divide x³ + x² + 1 by x - a by long division. Show that the quotient also has a zero in Z₂(a) by simply
trying the eight possible elements. Then complete the factorization.]
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