$\mathbb{R} [ X ] / ( X^ 2 + 2 )\cong \mathbb{C}$?

Question

I am puzzled since in my eyes I cannot prove that there can be a function which is homomorphic regarding multiplication. Do you know any functions that could be?

I was thinking that the quotient field has the format:

< ax+b | x^2 = -2>

Am I missing something?

Answer 1

Hint: Consider the map from $\mathbb R[x]$ into $\mathbb C$ defined by $p(x)\mapsto p\left(\sqrt 2\,i\right)$.

Answer 2

Define a ring homomorphism $\phi:\mathbb{R}[x]\to \mathbb{C}$ by $\phi(p(x))=p(i\sqrt{2})$. Then clearly $\phi $ is onto. Now it can be seen that $\mathbb{R}[x]$ is a principal ideal domain and its Ker$(\phi)=\langle x^2+2\rangle.$ Using fundamental theorem of ring homomorphism you will have the result.