field theory algebra 1. Show that F = Z₂[x] /< x³ + x + 1 > is a field and find its order.2. List the elements of F.3. Show that F* is cyclic group generated by [x+1].4. Write every non-zero element in F as powers of [x+1].5. Find the inverse of [x² + x + 1] in F.6. Find the sum of [x + 1]³ + [x + 1]5 + [x + 1]².7. List the subfields of F.

To show that F=Z2[x]⟨x3+x+1⟩ is a field, you need to demonstrate two key properties: that the…

Answered: 1. Show that F = Z₂[x] /< x³ + x+ 1 >…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.2: Divisibility And Greatest Common Divisor

Problem 33E

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1. Show that F = Z₂[x] /< x³ + x+ 1 > is a field and find its order. 2. List the elements of F. 3. Show that F* is cyclic group generated by [x+1]. 4 Write every non-zero element in F as powers of [x+11