Let $p$ and $q$ be (not necessarily distinct) odd primes and let $F=\mathbb{Q}_p(\mu_q)$. The $q^{th}$ Hilbert symbol induces a non-degenerate alternating form $$(\cdot,\cdot)_q:F^\times/(F^\times)^q\times F^\times/(F^\times)^q \to \mu_q,$$ which implies that there is a "Lagrangian decomposition" $L\oplus L^\vee=F^\times/(F^\times)^q$, where $L,L^\vee$ are subgroups such that the Hilbert symbol identifies $L^\vee$ with the Pontryagin dual of $L$, while being trivial if we restrict it to $L\times L$ or $L^\vee\times L^\vee$.
Local class field theory tells us that these subgroups are the norm subgroups of field extensions $F_L$ and $F_{L^\vee}$ of degree $\sqrt{|F^\times/(F^\times)^q|}$. My first question is
What is known about the fields $F_L$, or perhaps the pairs of fields $(F_L,F_{L^\vee})$, that arise in this way? Is there any literature on these pairs, particularly in the ramified setting?
Another question is how we can identify certain ``distinguished'' fields that arise this way.
Some basic examples, let's suppose $p\neq q$:
In this case, $F^\times/(F^\times)^q=\mathbb{F}_q^2$ and there is a distinguished Lagrangian: we can take $L=\mathcal{O}_F^\times/(\mathcal{O}_F^\times)^q$ to be the group of units mod $q$ powers. Then any choice of uniformizer $\pi\in F$ gives a complimentary group $L^\vee=\langle \pi \rangle$, and we have $$L\oplus L^\vee=F^\times/(F^\times)^q.$$ I say this is distinguished because the choice is uniform in $p\neq q$: we choose the kernel of the valuation character $x\mapsto \mathrm{val}_p(x)$. In this case, the field extension $F_L/F$ associated to the subgroup $L$ is the unique unramified extension of degree $q$, while $F_{L^\vee}=F(\sqrt[q]{\pi})$ is totally ramified.
Now let $p= q$:
In this case, $F^\times/(F^\times)^p\cong \mathbb{F}_p^{p+1}$ is much bigger, but we can (very naively!) hope to construct a Lagrangian in a "uniform way," that is as independent of $p$ as possible. This would correspond to a distinguished field extension $F_L$ of degree $p^{(p+1)/2}$.
Even for $p=3$, we are looking for degree $9$ fields, and by enumerating Lagrangian subspaces of $F^\times/(F^\times)^p\cong\mathbb{F}_3^4$, there are $$|\mathrm{Sp}_{4}(\mathbb{F}_3)/P_{Siegel}(\mathbb{F}_3)|=3^3+3^2+3+1=40$$ such fields, and I have no idea how to go about constructing all such fields.
So, here is my question:
Is there a uniform choice of Lagrangian in this ramified case? Equivalently, is there a distinguished degree $p^{(p+1)/2}$ extension of $\mathbb{Q}_p(\mu_p)$ whose norm subgroup contains the $p^{th}$ powers? For example, is there a natural linear map $\mathbb{F}_p^{p+1}\to\mathbb{F}_p^{(p+1)/2}$ I'm not seeing whose kernel gives the Lagrangian?
My motivation comes from automorphic forms, where the choice of such a Lagrangian (in a slightly more general setting) is useful for studying certain non-unique models of automorphic reps. In particular, it would be great to pin down choices at every finite place, including the ramified places.
Obviously, this question makes sense with $F$ any extension of $\mathbb{Q}_p(\mu_n)$, where $n$ can be any positive number. I would thrilled if there were a reference in this generality.
