It seems that padic polynomials simply have a malfunctioning _rmul_ !!!

    def _rmul_(self, left):
        return None
        # The code below has never been tested and is somehow subtly broken.

see:

https://github.com/sagemath/sage/blob/5188024881df509aee12e59e4d8ebf003c675e4a/src/sage/rings/polynomial/padics/polynomial_padic_capped_relative_dense.py#L576-L577

it shouldn't be too hard to fix that? Someone with some experience with the 
code there could probably fix that quite easily.

On Friday, 24 January 2025 at 13:38:27 UTC-8 Nils Bruin wrote:

> On https://github.com/sagemath/sage/pull/39371 I found that R['x'] knows 
> how to find the scalar multiplication action of R on R['x'] in most cases, 
> but that for R=Qp this is not the case. It looks like there are some 
> benefits from relying on such an action, so it would be nice if this action 
> were discoverable in all cases. My questions:
>  - where does this action get installed for most rings R?
>  - is there a reason why for Qp this action is not installed?
>
>

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