Let $\Bbb Q_p$ be a p-adic field. I know $\Bbb Q$ has infinitely many number of subrings since they are multiplicatively set of $\Bbb Z$ which are generated by primes.
Since $\Bbb Q$ is a subring of $\Bbb Q_p$, I understand $\Bbb Q_p$ has infinitely many number of subrings. But I cannot concretely describe all of them. Are there any known results?
Thank you for helping me.
On
Let $R\hookrightarrow \mathbb Q_p$ be an inclusion. We obviously have that $R$ is an integral domain, and hence induces a field extension $\mathbb Q_p/\text{Frac} R,$ where $\text{Frac} R$ denotes the quotient field of $R.$
Conversely, given a subfield $K\subseteq \mathbb Q_p,$ all integral domains $R$ with $\text{Frac} R=K$ induces an inclusion $R\subseteq \mathbb Q_p.$
My advice is: Forget about it – no real chance of classifying all the subrings.
Remember that $\Bbb{Q}_p$ is in a sense analogous to $\Bbb{R}$, and you don't really want to try to classify all the subrings of $\Bbb{R}$ either.
As Torsten Schoeneberg pointed out, $\Bbb{Q}_p$ has infinitely many distinct subfields. An easy way of seeing that is to observe that, due to Hensel lifting, $\Bbb{Q}_p$ contains the square roots of all the integers $n$ such that $n$ is a quadratic residue modulo $p$. That's infinitely many distinct quadratic subfields already let alone other algebraic extensions of $\Bbb{Q}$. I'm fairly sure that the transcendence degree of $\Bbb{Q}_p/\Bbb{Q}$ is also infinite (possibly continuum cardinality?). This really kills all hope.