Let a € Z2 be a zero of the irreducible polynomial x³ + x +1 over Z2.1. Show that the Frobenius automorphism ơ2 of Z2(@) permutes the elements ofA = {a, a², a + a²} and B = {1+ a, 1+ a², 1+ a + a²}.2. Verify that the elements of B are the zeros of x³ + x² + 1 in Z2.

The given problem is related to abstract algebra. Given: α ∈ ℤ2 is a zero of the irreducible…

Answered: Let a e Z2 be a zero of the 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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Let a e Z2 be a zero of the irreducible polynomial x³ + x +1 over Z2. 1. Show that the Frobenius automorphism ơ2 of Z2(a) permutes the elements of A = {a, a²,a+a²} and B = {1+a,1+a²,1+a+a²}. 2. Verify that the elements of B are the zeros of x³ + x² + 1 in Z2.