Question

[kwa...@chosun.ac.kr]

Dear the editor of LMFDB

My name is Kwang-Seob Kim, working at Chosun University.

I hope this message finds you well.

I am writing to inquire about a specific extension that I found in the LMFDB. My research concerns the field extensions over Q(√–5), and I noticed that when searching for extensions that are only ramified at the prime 5, the database shows exactly one field whose Galois closure over Q has a D₅ structure.

I have two main questions regarding this:

1. Does the data in LMFDB reflect that this D₅-extension is indeed unique under these conditions, or is it more accurate to state that it is the unique example within the current scope of the database?
2. Is there a possibility that there are additional fields satisfying these conditions which might not yet be included in LMFDB?

Any insights or further references you could provide on this matter would be greatly appreciated. My research involves the minimal ramification problem and class field theory, and a deeper understanding of this case would be invaluable.

Thank you very much for your time and assistance.

Best regards,
Kwang-Seob Kim.

[John Jones]

Hi,

A D_5 extension has a stem field in degree 5.  The page

  https://beta.lmfdb.org/NumberField/Completeness

the LMFDB contains all degree 5 extensions of Q unramified outside of {2,3,5,7}, so it contains all fields ramified only at 5.  The page describing various ways the LMFDB number field data is complete is accessible from the box to the upper right on any number field pages.

John Jones

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