I'm still preparing for my algebra test, and I ran into this problem:

"Let $R:=\{a+ib|a,b\in \mathbb{Q}\}$, a subring of $\mathbb{C}$. Show that ring $R$ is isomorphic to the quotient ring $\mathbb{Q}[x]/(x^2+1)$.

I know a theorem I might can use, but I have no idea how to apply it on the problem. The theorem states that:

First Isomorphism Theorem "Let $f:R\rightarrow S$ be a surjective homomorphism of rings with kernel $K$. Then the quotient ring $R/K$ is isomorphic to $S$."

I hope someone can help me, thank you.

  • $\begingroup$ Yes, define a suitable homomorphism $f\colon \mathbb{Q}[x]\rightarrow \mathbb{Q}(i)$. $\endgroup$ – Dietrich Burde Jan 20 '18 at 17:29

Hint: evaluation of polynomials in $\mathbb Q[x]$ at a fixed element in $\mathbb C$ creates a ring homomorphism into $\mathbb C$.

Pick the right thing to evaluate at, and check the image and kernel of the resulting map.


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