Question

8. List all irreducible polynomials with binary coefficients of degree 4 or less. (Hint: produce a...

8. List all irreducible polynomials with binary coefficients of degree 4 or less. (Hint: produce a times table that shows the minimum number of products needed.) Show these as binary numbers (omitting the indeterminant) and as decimal numbers (interpreting the binary number into decimal). Is 23 a prime polynomial in this field?

9. Interpreting these decimal numbers into coefficients of polynomials with binary coefficients, what is the product of 11 and 10 modulo 31 in GF(2^4) over P = 31? (Hint: it is not 17.) Use polynomial multiplication and reduction by long division.

10. Find a generator g for GF(2^4) over P=31. Show that g is a generator for the field. Find the powers of g that produce 10 and 11 (as in question 9) and show that their product using the sum of the generator's powers gives the same results you got in problem 9.

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