Defining an evaluation map between $\mathbb{Q}\lbrack x \rbrack$ and $\mathbb{Q}\lbrack x \rbrack/(x^2-5)$

Question

I need to define an evaluation mapping between $\mathbb{Q}\lbrack x \rbrack$ and $\mathbb{Q}\lbrack x \rbrack/(x^2-5)$. I know I want the identity to map to the identity, but I'm not sure what the identity of $\mathbb{Q}\lbrack x \rbrack/(x^2-5)$ is. I just need a starting point. Thanks.

Answer 1

If $R$ is a commutative ring, and $I$ is an ideal in $R$, then there exists a surjective homomorphism $\pi_{I}: R \to R/I$ , $\pi_{I}(r)=r+I$. In this case, take $R= \mathbb{Q}[x]$ and $I=(x^2-5)$. The identity of $R/I$ is $1_{R}+I=1+(x^2-5)$.