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

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.

• Canonical projection? – Thomas Shelby Feb 1 at 2:27

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 $$1_{R}+I=1+(x^2-5)$$.