Question: Let s(x) = x4 + x3 + 1 Z2[x]. (a) Prove that s(x) is irreducible. (b) What is the order of the field Z2[x]/(s(x))?

Let s(x) = x4 + x3 + 1 ∈ Z2[x].
(a) Prove that s(x) is irreducible.
(b) What is the order of the field Z2[x]/(s(x))?
(c) Find [x2 + x + l]-1 in Z2[x]/(s(x)). (Find a, b, c, d ∈ Z2 so that [x2 + x + 1] [ax3 + bx2 + cx + d] = [1].)
(d) Determine [x3 + x + l][x2 + 1] in Z2[x]/(s(x)).

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a s0 1 sl so sx has no root in Z 2 or linear factor in Z 2 x But perhaps we can factor sx ... View full answer

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