Let $p$ be a fixed prime number and denote $\mathbb Q_p$ the field of $p$-adic numbers.
For each positive integer $n$, I would like to construct a finite Galois extension $K/\mathbb Q$ of degree at least $n$ in which $p$ splits completely or, equivalently, such that $K$ embeds into $\mathbb Q_p$.
It is shown here that such extensions exist if one removes the assumption of being Galois.
Also, one guess would be the cyclotomic extensions of degree $m \geq n$. But there $p$ splits completely if and only if $p \equiv 1 \text{ mod } m$, which obviously implies $n \leq m < p$. So these do not give extensions of arbitrary large degree.
Another idea is to construct a polynomial $f \in \mathbb Z[X]$ irreducible over $\mathbb Q$ of degree $\geq n$ with the following two properties:
- For all roots $\eta$ of $f$, the field $\mathbb Q(\eta)$ is already a splitting field of $f$.
- There is one root of $f$ modulo $p$ that is a single root, e.g., $f$ is separable modulo $p$ and has a root modulo $p$.
By 2., we could then lift the single root modulo $p$ to a root $\eta \in \mathbb Q_p$ of $f$. By 1., $\mathbb Q(\eta) \subseteq \mathbb Q_p$ is a splitting field of a polynomial over $\mathbb Q$ and therefore Galois over $\mathbb Q$. For instance, 1. is satisfied if $f$ is the $q$-th cyclotomic polynomial, but that does not help us, as already noted above.
Thank you for your help!
