Generator of valuation ring for p-adic fields?

[Alex Braat]

Hello,

Is it maybe possible to include the generator of the valuation ring of p-adic fields in the database? 

Currently the generating polynomial for p-adic fields of non-prime degree does not give a generating element of the valuation ring. As the information is practically already on the page (one just has to construct the maximal unramified extension, then construct the relative Eisenstein extension on top of that and then compute the minimal polynomial of the sum of the generators), it would be a nice quality-of-life upgrade to include these.

I'm not entirely sure this is computationally feasible for the entire p-adic fields database and if you consider this to be worth the effort. Thoughts?

Best regards,

Alex Braat

[John Jones]

Hi,

I think this is a reasonable suggestion.  

Until it is implemented, here are a couple of suggestions.  I think all fields with residue field degree 1 are given by an Eisenstein polynomial, so it is more than fields of prime degree.  In all cases, the database does give the unramified/totally ramified tower for the field.

John Jones

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[Alex Braat]

Hi John,

Thank you for your response. 

While the database does give the tower, for my application I would like to download all p-adic fields into SageMath. As far as I know the "Download to SageMath" .sage file does not include the polynomials that define the tower. As p-adic extension that are not unramified or Eisenstein are not yet implemented into Sage, I have to manually copy the polynomials that define the tower for every extension. Maybe there is a smart work-around, but I have not found it yet.

Op maandag 8 maart 2021 om 15:39:15 UTC+1 schreef John Jones:

[Edgar Costa]

If you want, you can have access to the raw data via the API: https://www.lmfdb.org/api/

Cheers,

Edgar

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