A100980 Number of totally ramified extensions over Q_3 with degree n in the algebraic closure of Q_3.

1, 2, 21, 4, 5, 150, 7, 8, 5085, 10, 11, 2892, 13, 14, 10905, 16, 17, 984114, 19, 20, 137739, 22, 23, 472344, 25, 26, 900792441, 28, 29, 5314350, 31, 32, 17537487, 34, 35, 13832346276, 37, 38, 186535713, 40, 41, 602654010, 43, 44, 1408273477425, 46, 47, 6198727728, 49, 50, 19758444837, 52

Offset

1,2

Comments

a(n) is the number of 3-adic fields of the form Q_3(alpha) contained in some fixed algebraic closure A of Q_3, where alpha is a root of a 3-Eisenstein degree n polynomial over Q_3. Here, two fields Q_3(alpha) and Q_3(beta) are counted as distinct if they are distinct as subfields of A (even if they are Q_3-isomorphic fields). - Robin Visser, Nov 29 2025

References

M. Krasner, Le nombre des surcorps primitifs d'un degre donné et le nombre des surcorps métagaloisiens d'un degré donné d'un corps de nombres p-adiques. Comptes Rendus Hebdomadaires, Académie des Sciences, Paris 254, 255, 1962.

Jean-Pierre Serre, Une "formule de masse" pour les extensions totalement ramifiées de degré donné d'un corps local. C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 22, A1031-A1036.

Links

Robin Visser, Table of n, a(n) for n = 1..4000

Jordi Guàrdia-Rúbies, John W. Jones, Kevin Keating, Sebastian Pauli, David P. Roberts, and David Roe, Families of p-adic fields, arXiv:2507.02360 [math.NT], 2025.

Formula

a(n) = n*(Sum_{s=0..m} p^s*(p^(eps(s)*n)-p^(eps(s-1)*n))), where p=3, n=h*p^m, with gcd(h, p)=1, eps(-1)=-oo, eps(0)=0 and eps(s)=Sum_{i=1..s} 1/(p^i).

Example

a(4)=4: There are 4 totally ramified extensions all with Galois group D_8, so two of them are isomorphic to Q_3[x]/(x^4+3) and two of them are isomorphic to Q_3[x]/(x^4-3).

Maple

a := proc(n) local s, nn, i, j; if n mod 3 <> 0 then return n fi;
 s, nn := 0, n;
 while nn mod 3 = 0 do s:=s+1; nn:=nn/3; od; n*(1+3*(3^(n/3)-1)
 + add(3^(s+add(n/3^j, j=1..s))-3^(s+add(n/3^j, j=1..s-1)), i=2..s));
end; seq(a(i), i=1..20); # Brendan McKay, Feb 25 2026

Prog

(SageMath)
def a(n):
    if n%3 != 0: return n
    s, ans = Integer(n).valuation(3), 1 + 3*(3^(n/3)-1)
    for i in range(2, s+1):
        ans += 3^(i+sum([n/(3^j) for j in range(1, i+1)]))
        ans -= 3^(i+sum([n/(3^j) for j in range(1, i)]))
    return n*ans  # Robin Visser, Nov 29 2025

Crossrefs

Cf. A100976, A100977, A100978, A100979, A100981, A100983, A100984, A100985, A100986.

Sequence in context: A303218 A162536 A377413 * A360980 A122509 A024230

Adjacent sequences: A100977 A100978 A100979 * A100981 A100982 A100983

Keyword

nonn

Author

Volker Schmitt (clamsi(AT)gmx.net), Nov 25 2004

Status

approved