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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

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

Advanced Engineering Mathematics
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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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.
Transcribed Image Text: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.
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