What does the vector space $\mathbb{Q}[z]/\mathbb{Q}[z]·P(z)$ of the polynomials modulo the multiples of $P(z)$ represent? I only wish to understand a sentence which is unclear to me. It's the following:
Let $P(z) = z^2 − z − 56 \in \mathbb{Q}[z]$, $V$ be the $\mathbb{Q}$-vector space $\mathbb{Q}[z]/\mathbb{Q}[z]·P(z)$ of the polynomials modulo the multiples of $P(z)$.
What does the space $V$ actually represents? For instance, how can we find the images of very elmentary polynomials like $1$ or $z$ in $V$? I am confused with the quotient multiplied by multiples of $P(z)$.
 A: First of all note that $P(z)$ is irreducible in $\mathbb{Z}[z]$, because it is of degree $2$ and does not have a root in $\mathbb{Z}$. Then it is also irreducible in $\mathbb{Q}[z]$, so the ideal generated by it is a maximal ideal. If we consider a root $\alpha\in\mathbb{C}$ of $P$, we can look at the evaluation homomorphism $ev_\alpha:\mathbb{Q}[z]\rightarrow \mathbb{C}$, that takes a polynomial in $\mathbb{Q}[z]$ and evaluates it at $z=\alpha$. Since $\alpha$ is a root of $P$, we have that the ideal generated by $P$ is in the kernel, hence we have equality since the ideal generated by $P$ is maximal. Then the isomorphism theorem gives us $$\mathbb{Q}[z]/(P)=\mathbb{Q}[z]/\operatorname{ker}(ev_{\alpha})\cong \operatorname{im}(ev_\alpha)=\mathbb{Q}[a]\subseteq\mathbb{C}.$$ In conclusion, we can view this quiotent as adjoining a root of $P$ to $\mathbb{Q}$; this space has dimension $2$ over $\mathbb{Q}$. We can see this using the uniqueness of the division with remainder in $\mathbb{Q}[z]$. A $\mathbb{Q}$-basis is then seen to be $1,z$ (the element $z^2$gets identified with $z+56$ in the quotient), and via $ev_\alpha$ we see that a $\mathbb{Q}$-basis of $\mathbb{Q}[\alpha]$ is given by $1,\alpha$.
