Question #137503
prove that Q[x]/<x-2> is isomorphic to Q as fields.
1
Expert's answer
2020-10-11T17:54:07-0400

Polynomial remainder theorem implies that each element of Q[x]/x2\mathbb Q[x]/\langle x-2\rangle is of the form [f(x)]=f(x)+x2=f(x)+(x2)Q[x][f(x)]=f(x)+\langle x-2\rangle=f(x)+(x-2)\mathbb Q[x], where degf(x)<deg(x2)=1\deg f(x)<\deg(x-2)=1, and therefore, f(x)Qf(x)\in\mathbb Q. Consequently, Q[x]/x2={[a] : aQ}\mathbb Q[x]/\langle x-2\rangle=\{[a]\ : \ a\in\mathbb Q\}.


Consider the map ψ:QQ[x]/x2,ψ(a)=[a].\psi: \mathbb Q\to \mathbb Q[x]/\langle x-2\rangle, \psi(a)=[a]. Taking into account that ψ(ab)=[ab]=[a][b]=ψ(a)ψ(b)\psi(ab)=[ab]=[a][b]=\psi(a)\psi(b) and ψ(a+b)=[a+b]=[a]+[b]=ψ(a)+ψ(b)\psi(a+b)=[a+b]=[a]+[b]=\psi(a)+\psi(b), we conclude that ψ\psi is a field homomorphism. If aba\ne b, then ψ(a)=[a][b]=ψ(b)\psi(a)=[a]\ne [b]=\psi(b), and consequently ψ\psi is injective. Since ψ(a)=[a]\psi(a)=[a] for any [a]Q[x]/x2[a]\in \mathbb Q[x]/\langle x-2\rangle, ψ\psi is surjective. Therefore, ψ\psi is a field isomorphism. Thus the fields Q\mathbb Q and Q[x]/x2\mathbb Q[x]/\langle x-2\rangle are isomorphic.


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