I am studying the proof of Proposition 1 and I don't understand the following two statements.
As $p\mid\Delta_K$ the group $(\mathcal{O}_{\Delta_K}/p\mathcal{O}_{\Delta_K})^*$ is isomorphic to $(\mathbb{F}_p[X]/(X^2))^*$. This group contains $p(p-1)$ elements of the form $a+b\sqrt{\Delta_K}$ where $a\in (Z/pZ)^*$ and $b\in Z/pZ$
Here, $\Delta_K=-p$ and $\Delta_K\equiv1\pmod4$.
How did they come up with the isomorphism and why does the latter contain elements of the form $a+b\sqrt{\Delta_K}$? I do not have a background in group/ring theory and hence the question.