Question: 5. (30 points) Generate the multiplication table for the extension field GF(2) for the case that the irreducible polynomial is P(x) = 23 + +

5. (30 points) Generate the multiplication table for the extension field GF(2") for the case that the irreducible polynomial is P(x) = 23 + + 1. The multiplication table is in this case a 8 x 8 table. hint: Start by listing all the elements of GF(23) then review the readings about fields and groups. Review the example that was done in class. Review the xtimes method for multiplication from YouTube on the weekly page. The table below has some sample entries to make this easier. Use the actual elements, not GF(n) notation. 10 1 x GF(3) |GF(4) |GF(5) | GF(6) | x2 + + 1 GF(0) GF(1) GF(2) GF(3) 0 r+1 GF(4) GF(5) GF(6) GF(7) Table 1: One way to report your results is as a multiplication table for the extension field GF(23) A second alternative way of presenting your results is as an orderly list, as shown below. . 0* 0 =0 . 0*1 = 0 . (x' +x+ 1) * (x' +x+ 1)= (answer here)

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