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The polyromial X+X+1 is ireducilole aver Z2. Use it to Canstruct additlon and multiplcotion table of a field with 8 elements

The polyromial X+X+1 is ireducilole aver Z2. Use it to Canstruct additlon and multiplcotion table of a field with 8 elements

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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The polyramral XX+1 is irreduciloe aer Z2. Use it to Canstruct additlon and multiplloation table of a field with 8 elements
Transcribed Image Text:The polyramral XX+1 is irreduciloe aer Z2. Use it to Canstruct additlon and multiplloation table of a field with 8 elements
Expert Solution
Step 1

Let F8 be a finite field with 8 elements.

Since the polynomial x3+x+1 is irreducible in ℤ2x, therefore, the quotient ring ℤ2xx3+x+1 is a field.

The order of the field  ℤ2xx3+x+1 is 23, that is, 8.

Hence, F8 and ℤ2xx3+x+1 are isomorphic, that is,

                       F8≅ℤ2xx3+x+1

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