Automorphisms of $\mathbb C_p$ with constraints

In Automorphisms of $\mathbb C_p$, K. Conrad showed that there exist uncountably many $$\mathbb Q_p$$-automorphisms of $$\mathbb C_p$$. I have a quite similar question.

Let $$(a_n)_{n\in\mathbb N}$$ and $$(b_n)_{n\in\mathbb N}$$ be sequences in $$\mathbb Q_p$$. Does there exist an uncountable set $$S$$ of continuous automorphisms of $$\mathbb C_p$$ that leave invariant $$\{b_n\mid n\in\mathbb N\}$$ and such that the set of images of the sequence $$(a_n)_{n\in\mathbb N}$$ by the elements of $$S$$ is uncountable. I assume that for all $$n\in\mathbb N$$, $$b_n\notin\mathbb Q(\smash{a_j}_{\mid j\in\mathbb N})$$.

• I do not think this is a "quite similar question" to your previous one. Here you want $S$ to contain continuous automorphisms of $\mathbf C_p$. If you mean they are continuous using the ordinary topology on $\mathbf C_p$ then they fix all elements of $\mathbf Q_p$ (they fix all elements of $\mathbf Q$ by algebra and then all of $\mathbf Q_p$ by continuity) so the image of each $a_n$ by $S$ is just $a_n$. If you did not want elements of $S$ to be continuous for the ordinary topology on $\mathbf C_p$ then what topology did you have in mind? – KConrad Mar 13 '19 at 8:12
• By the way the cardinal of the automorphism group of $\mathbf{C}_p$ as $\mathbf{Q}_p$-algebra is not only uncountable, but $2^{2^{\aleph_0}}$. Its subgroup of continuous automorphisms has a natural topology of infinite second countable profinite group, and has cardinal $2^{\aleph_0}$. – YCor Mar 13 '19 at 9:51
• By the way, the word is 'automorphism' (ending in 'ism'), not 'automorphim' (ending in 'im'). I have edited to correct. – LSpice Mar 13 '19 at 17:02
• The continuous automorphism group is $\text{Gal}(\overline{\mathbf{Q}_p}/\mathbf{Q}_p)$ the easy part is $T = \lim_{n \to \infty} \mathbf{Q}_p(\zeta_{p^{n!}-1}, p^{1/(p^{n!}-1)})$ the maximal tamely ramified extension whose Galois group is $\prod_{\ell \ne p} \text{Aff}(\mathbf{Z}_{\ell})$ and the complicated part is the remaining tower of totally ramified extensions – reuns Mar 13 '19 at 19:30