I am trying to prove that the following objects are isomorphic: $$\mathbb{Q}[x]/<x+1> \cong \mathbb{Q}$$

but I am not sure where to start. Is finding an actual map from one to another the way to approach this? I would appreciate any hint on this.

  • 1
    $\begingroup$ That makes no sense: $\mathbf Q[x]/(x+1)$ looks like a quotient ring. Do you mean you are trying to prove the principal ideal $(x+1)$ in $\mathbf Q[x]$ is isomorphic to the princiapl ideal $(1) = \mathbf Q[x]$? If so, what do you mean by "isomorphism of ideals"? $\endgroup$ – KCd 2 days ago
  • $\begingroup$ There is a slightly easier way in that you don't actually have to deal with cosets and quotients directly. You still find a map, but here you find a map $\phi:\mathbb{Q}[x]\to \mathbb{Q}$ which is surjective and where $\ker \phi = (x+1)$. Then we would have that $\mathbb{Q}[x]/(x+1) = \mathbb{Q}[x]/\ker \phi \cong \operatorname{im} \phi = \mathbb{Q}$ by one of the isomorphism theorems. $\endgroup$ – memerson 2 days ago
  • $\begingroup$ Suppose $ a + \langle x+1 \rangle \in \mathbb{Q}/\langle x + 1 \rangle$. Now show that $a$ must be a rational number. I hope it's easy to proceed now. $\endgroup$ – Strange 2 days ago

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