Right thanks, my mistake

From Hensel’s lemma there’s a primitive 8th root of unity in $K$ if there’s one in the residue field of $K$

The isomorphism between Galois groups of the field extension and the residue field extension is canonical, so there’s a canonical generator corresponding to frobenius

Yes— the exterior derivative (and differential forms in general) can be justified as the result of adding “correction factors” to familiar constructions from calculus in order to make them coordinate invariant.

@NoahB I don't know, but if you know how to show that I would be very interested to see it

This is a guess, but I think it's a sort of "tensor". You have groups $H \subseteq G$ and $H$ acting on a set $X$. The object you want is the Cartesian product $G \times X$ modulo the relations $(gh, x) \sim (g, hx)$ for all $g, h, x$. The $G$–action acts on the first coordinate.

One reason may be that the structure of multiplicative groups of local fields is very explicit (it’s on wikipedia). Just looking at it , in most cases I don’t think there are many interesting continuous homomorphisms from the multiplicative group of a local field $K$ to $\mathbb{Q}_p$, especially when $p$ differs from the residue characteristic of $K$. Also, morphisms from $K^\times$ to $\mathbb{C}$ include all morphisms to $\mathbb{R}$ already.

@Boris math.wustl.edu/~kumar/papers/seshadri.pdf i have a feeling the intended result is here

Great answer, thank you

Here’s a question I am not qualified to answer. Topological vector bundles are all pullbacks of the tautological bundle on the classifying space. Can we give a geometric intepretation of the tensor product using this?

For me the intuition is just that tensor product is analogous to multiplication of functions. Note that when tensoring line bundles on a scheme, the associated divisors add, the same as they do for functions. Consider the group of fractional ideals in a Dedekind domain: the ideals are not strictly speaking numbers, but passing to some extension of the fraction field any such fixed ideal becomes principal. The fractional ideals are directly analogous to line bundles on Riemann surfaces.