Uniquely orderable subfields of $\mathbb{Q}_p$?

Question

I have heard that, unlike $\mathbb{R}$, the field $\mathbb{Q}_p$ cannot be realised an ordered field. Is there any way to extend the natural ordering on $\mathbb{Q}$ to a larger subfield of $\mathbb{Q}_p$ in a natural way?

Answer 1

In an answer to your previous question Are $\mathbb{R}$ and $\mathbb{Q}$ the only subfields of $\mathbb{C}$ with natural structure as ordered fields? it is mentioned that $\mathbb Q(\sqrt[3]{2})$ has a unique field ordering. The same holds true for any $\mathbb Q(\sqrt[n]{q}) \simeq \mathbb Q[X]/(X^n-q)$ for $n$ an odd integer $\ge 3$ and $q$ a positive rational number which is not an $n$-th power of a rational.

Now Hensel's Lemma will show that any given $\mathbb Q_p$ contains many subfields of that form. E.g. $\mathbb Q_2$ contains $\mathbb Q(\sqrt[3]{3})$, both $\mathbb Q_5$ and $\mathbb Q_7$ contain $\mathbb Q(\sqrt[3]{2})$, $\mathbb Q_{11}$ contains $\mathbb Q(\sqrt[3]{5})$ ...