A100978 Number of all extensions over Q_5 with degree n in the algebraic closure of Q_5.

1, 3, 4, 7, 106, 12, 8, 15, 13, 1818, 12, 28, 14, 24, 12424, 31, 18, 39, 20, 109242, 32, 36, 24, 60, 8281131, 42, 40, 56, 30, 4687272, 32, 63, 48, 54, 15624848, 91, 38, 60, 56, 146484090, 42, 96, 44, 84, 634765378, 72, 48, 124, 57

Offset: 1

Author

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

Keywords

• nonn

Examples

			a(2)=3 There are 2 ramified extensions with minimal polynomials x^2-5, x^2-10 and one unramified x^2+4*x+2.
		

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, A100979, A100980, A100981, 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^(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;

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=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)