Abstract Algebra: Theory and Applications · Problem 21.5.5

Finite Fields — Thomas Judson

Mathematics — Abstract Algebra · finite-field-quotient-ring

Problem

Show that Z2[x]/⟨x3+x+1⟩\mathbb{Z}_2[x]/\langle x^3 + x + 1\rangle is a field with eight elements. Construct a multiplication table for the multiplicative group of the field. Hint. Use the fact that the elements of Z2[x]/⟨x3+x+1⟩\mathbb{Z}_2[x]/\langle x^3 + x + 1\rangle are 0,1,α,1+α,α2,1+α2,α+α2,1+α+α20, 1, \alpha, 1+\alpha, \alpha^2, 1+\alpha^2, \alpha+\alpha^2, 1+\alpha+\alpha^2 and the fact that α3+α+1=0\alpha^3 + \alpha + 1 = 0.

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