A100977 Number of all extensions over Q_3 with degree n in the algebraic closure of Q_3.

1, 3, 22, 7, 6, 228, 8, 15, 5323, 18, 12, 5068, 14, 24, 13092, 31, 18, 1495839, 20, 42, 157424, 36, 24, 885660, 31, 42, 942953404, 56, 30, 9565848, 32, 63, 19131816, 54, 48, 24240086731, 38, 60, 200884628, 90, 42, 1033121184, 44, 84

Offset

1,2

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.

Links

Table of n, a(n) for n=1..44.

Formula

a(n)=(sum_{d|h}d)*(sum_{s=0}^m (p^(m+s+1)-p^(2*s))/(p-1)*(p^(eps(s)*n)-p^(eps(s-1)*n))), where p=3, n=h*p^m, with gcd(h, p)=1, eps(-1)=-infinity, eps(0)=0 and eps(s)=sum_{i=1 to s} 1/(p^i)

Example

a(2)=3 There are 2 ramified extensions with minimal polynomials x^2+3, x^2-3 and one unramified x^2+2*x+2.

Maple

p:=3; eps:=proc()local p, s, i, sum; p:=args[1]; s:=args[2]; if s=-1 then return -infinity; fi; if s=0 then return 0; fi; sum:=0; for i from 1 to s do sum:=sum+1/p^i; od; return sum; end: ppart:=proc() local p, n; p:=args[1]; n:=args[2]; return igcd(n, p^n); end: qpart:=proc() local p, n; p:=args[1]; n:=args[2]; return n/igcd(n, p^n); end: logp:=proc() local p, pp; p:=args[1]; pp:=args[2]; if op(ifactors(pp))[2]=[] then return 0; else return op(op(ifactors(pp))[2])[2]; fi; end: summe:=0; m:=logp(p, ppart(p, n)); h:=qpart(p, n); for s from 0 to m do summe:=summe+(p^(m+s+1)-p^(2*s))/(p-1)*(p^(eps(p, s)*n)-p^(eps(p, s-1)*n)); od; a(n):=sigma(h)*summe;

Crossrefs

Cf. A100976, A100978, A100979, A100980, A100981, A100983, A100984, A100985, A100986.

Sequence in context: A016449 A161377 A122495 * A364163 A037101 A226028

Adjacent sequences: A100974 A100975 A100976 * A100978 A100979 A100980

Keyword

nonn

Author

Volker Schmitt (clamsi(AT)gmx.net), Nov 24 2004

Status

approved