Auume we know that m(X) = X4 + X + 1 ∈ Z2[X] is irreducible. Let K denote the field Z2[X]/(m(X)) and let α denote [X]m(X) ∈ K.For b1, b2, . . . , bk ∈ K prove the identity (b1 + b2 + . . . + bk)2=b12 + b22 + . . . + bk2Find the roots of m(X) in K. Express these roots in terms of α. (α ∈ K is a root of m(X). Hint: Please find the rest of the roots or to prove that there are no more roots)Factor m(X) into irreducible polynomials in K[X].

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Q: + Bz³) ûx + y)ûz

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