A100981 Number of totally ramified extensions over Q_5 with degree n in the algebraic closure of Q_5.

1, 2, 3, 4, 105, 6, 7, 8, 9, 1210, 11, 12, 13, 14, 9315, 16, 17, 18, 19, 62420, 21, 22, 23, 24, 8203025, 26, 27, 28, 29, 2343630, 31, 32, 33, 34, 13671735, 36, 37, 38, 39, 78124840, 41, 42, 43, 44, 439452945, 46, 47, 48, 49, 295410156050, 51

Offset: 1

Author

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

Keywords

• nonn

Examples

			a(3)=3: there is one totally ramified extension with Galois group S_3, so there are 3 totally ramified extensions in the algebraic closure all isomorphic to Q_5[x]/(x^3+5)
		

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, A100980, A100983, A100984, A100985, A100986.

Programs

Maple

p:=5; 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=5, 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)