I know the following fact from basic number theory.
Let $K=\Bbb{Q}(\sqrt{d})$ be a quadratic number field. Let $p$ be a prime. Then the fact that there is only one prime $\mathfrak{P}$ above $p$ in $K$ is the same thing as saying $(\frac{d}{p})\neq 1$. The symbol is the well known Legendre symbol.
I am wondering if there exists a statement similar to this when $\Bbb{Q}$ is replaced by any number field $F$. That is, let us consider the extension $F(\sqrt{d})$ over $F$. Suppose $P$ is a prime ideal of $F$. Is it true that:
There exists only one prime of $F(\sqrt{d})$ over $P$ iff $(\frac{d}{P})\neq 1$ with some meaning attached to $(\frac{d}{P})$.
Any comments in this direction will be helpful. Thanks.
You can define $\left(\frac{d}{P}\right)$ to be $1$ if and only if $d\notin P$ and $d$ is a square modulo $P$. Then the answer to your question is yes. In fact the following very useful theorem holds:
Let $F/K$ be an extension of number fields and let $\mathcal O_F$ and $\mathcal O_K$ be their number rings. Suppose $F=K(\alpha)$ for some $\alpha\in F$ and let $P\subseteq \mathcal O_K$ be a prime ideal such that $p\nmid [\mathcal O_F\colon \mathbb Z[\alpha]]$, where $p$ is the rational prime lying under $P$. Let $f$ be the minimal polynomial of $\alpha$ over $K$. Suppose that $f$ decomposes as $\overline{g_1}(x)^{e_1}\cdot\ldots\cdot \overline{g_s}(x)^{e_s}$ in $(\mathcal O_K/P)[x]$, where the $g_i(x)$ are monic polynomials in $\mathcal O_K[x]$ and the bar means the image in the quotient ring. Then the decompositon of $P\mathcal O_F$ is given by $Q_1^{e_1}\cdot\ldots\cdot Q_s^{e_s}$, where $Q_i=P\mathcal O_F+g_i(\alpha)$. Moreover, the inertia degree of $Q_i$ coincides with the degree of $g_i$.
Now it should be easy to apply this theorem in your situation!
If you want a more conceptual explanation, an unramified prime splits in a quadratic extension precisely when its Frobenius is trivial. Then the quantity you are looking at is precisely the Artin symbol of $P$, see https://en.wikipedia.org/wiki/Artin_reciprocity_law for details.