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Define a relation on Z2[x] by f ∼ g if g - f is a multiple of the polynomial x2 + x + 1

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Define a relation on Z2[x] by f ∼ g if g - f is a multiple of the polynomial x² + x + 1.

a) Show that ∼ is an equivalence relation.

For f ∈ Z₂[x], write [f] for its equivalence class. Let F denote the set of all of the equivalence classes.

b) How many equivalence classes are there? [Hint: what are the possible remainders when you divide by x 2 + x + 1?]

c) Show that defining [f] + [g] = [f + g] and [f] · [g] = [fg] gives a well-defined pair of binary operations on F.

d) Prove that the multiplication operation is associative. [You may assume that multiplication on Z2[x] is associative. The proof that F satisfies commutativity, distributivity etc. should be similar and is omitted.]

e) Show that F is a field