Add nicknames for unramified and tame p-adic fields (LMFDB#6201)#26
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Add nicknames for unramified and tame p-adic fields (LMFDB#6201)#26roed-math wants to merge 1 commit into
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Only quadratic p-adic fields had a readable nickname (Q_p(sqrt d)).
Nickname every unramified extension as Q_p(zeta_{p^f-1}) and every tame
extension as Q_p(zeta_{p^f-1}, root[e](zeta^k p)); the f=1 radicand is
written p*r to match the existing sqrt(p*u) convention, with k=0 and
zeta^k=-1 simplified. The twist exponent k is read from the stored
Eisenstein/unramified data (no discrete log) and normalized to a canonical
Frobenius-orbit representative so Q_p-isomorphic presentations coincide;
wild fields keep their label. Nicknames now show on field homepages, the
field-data knowl, subfield/completion displays, and the families Base column.
Verified with sage/pari over all 7731 tame degree>2 fields: 0 mismatches
vs an independent discrete-log computation, per-(p,e,f) counts equal the
number of Frobenius orbits, and each nickname field is Q_p-isomorphic to
the stored field. pytest local_fields: 9 passed (new test_nicknames);
pyflakes clean.
Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
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Previously only quadratic p-adic fields got a readable nickname (
Q_p(sqrt d)). This adds nicknames forevery unramified extension, written
Q_p(zeta_{p^f-1}), and every tamely ramified extension, writtenQ_p(zeta_{p^f-1}, root[e](zeta^k p))(with thef=1radicand written as an integerp*rto match theexisting
sqrt(p*u)convention, and simplifications fork=0andzeta^k=-1). The twist exponentkiscomputed directly from the stored Eisenstein and unramified data without a discrete log and normalized to a
canonical Frobenius-orbit representative, so Q_p-isomorphic presentations coincide; wildly ramified fields
keep their label. Nicknames now appear on field homepages (title and unramified-subfield line), the
field-data knowl, subfield/completion displays, and the families "Base" column. Verified against sage/pari
over all 7731 tamely-ramified fields of degree > 2: zero mismatches against an independent discrete-log
computation, per-(p,e,f) counts equal to the number of Frobenius orbits, and each nickname field confirmed
Q_p-isomorphic to the stored field. Addresses LMFDB#6201.
🤖 Generated with Claude Code