Let $\Bbb Z_p$ be the ring of p-adic integers. Then, I can list some subrings of $\Bbb Z_p$, like $\Bbb Z$, and... localization of $\Bbb Z$ at $(p^n)$ for some nonnegative integer $n$.
So I can say the number of subrings of $\Bbb Z_p$ is infinite, but I cannot concretely describe all of them. Are there any known results?
I would be appreciated if you help me describing all of subrings of $\Bbb Z_p$. Thank you.
There are infinitely many subrings of $\mathbb Z_p$ of the form $\mathbb Z[\alpha]$ for $\alpha$ algebraic over $\mathbb Q$, of any degrees you like, via Hensel's lemma. For example, for $p=1\mod 4$, there is $\alpha\in\mathbb Z_p$ such that $\alpha^2=-1$. Generally, for an algebraic extension $\mathbb Q(\alpha)$ (with $\alpha$ integral over $\mathbb Z$) in which $p$ splits completely, $\mathbb Z_p$ contains a copy of the algebraic integers of the extension.
Also, on cardinality considerations, there are uncountably many $\alpha\in\mathbb Z_p$ that are transcendental over $\mathbb Q$.
I doubt that all subrings are classifiable...