A100979 Number of totally ramified extensions over Q_2 with degree n in the algebraic closure of Q_2.

1, 6, 3, 92, 5, 90, 7, 5880, 9, 630, 11, 23028, 13, 3570, 15, 6021104, 17, 18414, 19, 2580460, 21, 90090, 23, 377290728, 25, 425958, 27, 233963492, 29, 1966050, 31, 1578396286944, 33, 8912862, 35, 19308478428, 37, 39845850, 39, 10823233699800, 41, 176160726, 43, 1511459389396, 45

Offset

1,2

Comments

a(n) is the number of 2-adic fields of the form Q_2(alpha) contained in some fixed algebraic closure A of Q_2, where alpha is a root of a 2-Eisenstein degree n polynomial over Q_2. Here, two fields Q_2(alpha) and Q_2(beta) are counted as distinct if they are distinct as subfields of A (even if they are Q_2-isomorphic fields). - Robin Visser, Nov 29 2025

References

M. Krasner, Le nombre des surcorps primitifs d'un degré 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..3000

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=2, n=h*p^m, with gcd(h, p)=1, eps(-1)=-infinity, eps(0)=0 and eps(s)=Sum_{i=1..s} 1/(p^i).

Example

a(2)=6: There are 6 ramified extensions with minimal polynomials x^2+2, x^2-2, x^2+6, x^2-6, x^2+2x+2, x^2+2x+6. There is another one by x^2+x+1, but this is unramified.

Maple

a := proc(n) local s, nn, i, j; if n mod 2 = 1 then return n fi; s, nn := 0, n;
 while type(nn, `even`) do s:=s+1; nn:=nn/2; od;
  n*(1+2*(2^(n/2)-1)
  + add(2^(s+add(n/2^j, j=1..s))-2^(s+add(n/2^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%2 != 0: return n
    s, ans = Integer(n).valuation(2), 1 + 2*(2^(n/2)-1)
    for i in range(2, s+1):
        ans += 2^(i+sum([n/(2^j) for j in range(1, i+1)]))
        ans -= 2^(i+sum([n/(2^j) for j in range(1, i)]))
    return n*ans  # Robin Visser, Nov 29 2025

Crossrefs

Cf. A100976, A100977, A100978, A100980, A100981, A100983, A100984, A100985, A100986.

Sequence in context: A367377 A324004 A002610 * A279107 A266661 A136133

Adjacent sequences: A100976 A100977 A100978 * A100980 A100981 A100982

Keyword

nonn

Author

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

Extensions

Corrected a(40) and more terms from Robin Visser, Nov 29 2025

Status

approved