Polynomial remainder theorem implies that each element of Q[x]/⟨x−2⟩ is of the form [f(x)]=f(x)+⟨x−2⟩=f(x)+(x−2)Q[x], where degf(x)<deg(x−2)=1, and therefore, f(x)∈Q. Consequently, Q[x]/⟨x−2⟩={[a] : a∈Q}.
Consider the map ψ:Q→Q[x]/⟨x−2⟩,ψ(a)=[a]. Taking into account that ψ(ab)=[ab]=[a][b]=ψ(a)ψ(b) and ψ(a+b)=[a+b]=[a]+[b]=ψ(a)+ψ(b), we conclude that ψ is a field homomorphism. If a=b, then ψ(a)=[a]=[b]=ψ(b), and consequently ψ is injective. Since ψ(a)=[a] for any [a]∈Q[x]/⟨x−2⟩, ψ is surjective. Therefore, ψ is a field isomorphism. Thus the fields Q and Q[x]/⟨x−2⟩ are isomorphic.
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