details of proof showing when (p) remains prime/is inertial This is from Ireland and Rosen's A Classical Introduction to Modern number theory and regards the proof to proposition 13.1.3.
background: Let $D$ be the ring of integers in a quadratic number field and P a prime ideal of $D$. The quadratic number field equal to $Q(\sqrt{d})$ for some square free integer $d$.
Assume degree of $P$ is 1 so that there is a bijection between $\mathbb{Z}/p \mathbb{Z} $ and $D/P$ where p is a prime in $P$. Then we can find an integer 'a' $ \in$ $\mathbb{Z}$ such that $a \equiv \sqrt{d}$ $  P$.
How can we guarantee we can find such 'a' to make the modular equation valid?
thanks
 A: I mean, it depends on what you know. 
Assume for a second that $d\not\equiv 1\mod 4$, so that $D=\mathbb{Z}[\sqrt{d}]$.
It's a common fact then that if we factor $x^2-d$ in $\mathbb{F}_p[x]$, say as $\overline{f_1(x)}^{e_1}\cdots \overline{f_m(x)}^{e_m}$ (where $f_i(x)\in\mathbb{Z}[x]$ and $\overline{f_i(x)}$ are distinct irreducibles) then $pD$ factors as $M_1^{e_1}\cdots M_m^{e_m}$ where $M_i=(p,f_i(\sqrt{d}))$. (this is called variably Dedekind's theorem or the Dedekind-Kummer theorem)
From this it's easy to deduce that the inertial degree of $M_i/(p)$ is $\deg f_i$. So, for example,  if we know that there is some prime of interial degree $1$ over $p$, then either $x^2-d$ ramifies mod $p$ as $(x-\zeta)^2$ for some $\zeta\in\mathbb{F}_p$ or splits completely as $(x-\zeta)(x+\zeta)$. Either way, we see that $x^2-d$ splits in $\mathbb{F}_p$.
EDIT: Note that $D$ is just $\mathbb{Z}[x]/(x^2-d)$. Suppose that $x^2-d$ has no solutions in $\mathbb{F}_p$. Note then that 
$$D/pD\cong \mathbb{Z}[x]/(x^2-d,p)\cong \mathbb{F}_p[x]/(x^2-d)$$
which is an integral domain. Thus, we see that $p\mathbb{Z}[\sqrt{d}]$ is prime, and since $p\mathbb{Z}[\sqrt{d}]\subseteq P$ this implies that $P=p\mathbb{Z}[\sqrt{d}]$. But, this implies that the inertial degree of $P$ over $p$ is $2$, and not one. 
So, if the inertial degree of $P$ over $p$ is $1$, then $x^2-d$ has a solution in $\mathbb{F}_p$.
