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

Offset: 1

Author

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

Keywords

• nonn

Examples

			a(4)=4 There are 4 totally ramified extensions both with Galoisgroup D_8, so 2 of them are isomorphic to Q_3[x]/(x^4+3) and two of them are isomorphic to Q_3[x]/(x^4-3)
		

References

• M. Krasner, Le nombre des surcorps primitifs d'un degre donne et le nombre des surcorps metagaloisiens d'un degre donne d'un corps de nombres p-adiques. Comptes Rendus Hebdomadaires, Academie des Sciences, Paris 254, 255, 1962

Crossrefs

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

Programs

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^s*(p^(eps(p,s)*n)-p^(eps(p,s-1)*n)); od; a(n):=n*summe;

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)=-infinity, eps(0)=0 and eps(s)=sum_{i=1 to s} 1/(p^i)