If there is an embedding from K to the p-adic numbers then p splits in K

Question

I am reading Serge Lang's book on elliptic functions, chapter 13, and this came up:

let $k$ be a quadratic number field given with an embedding $k \rightarrow Q_p$. Then $p$ splits in $k$.

I dont understand where this comes from

Answer 1

Outline: $K\hookrightarrow\mathbb Q_p$ then in particular $\mathcal O_K\subseteq \mathbb Z_p$. Now the kernel of the composition $\mathcal O_K\hookrightarrow\mathbb Z_p\to \mathbb F_p$ is a prime ideal $\mathfrak p$ of $\mathcal O_K$ dividing $(p)$. Checking $\mathfrak p\ne(p)$ is equivalent to checking $\mathcal O_K/p\ne\mathbb F_p$. But that's true since $\mathcal O_K$ is a free abelian group of rank $2$.

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