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%I #9 Jan 09 2013 16:43:28
%S 1,2,3,4,105,6,7,8,9,1210,11,12,13,14,9315,16,17,18,19,62420,21,22,23,
%T 24,8203025,26,27,28,29,2343630,31,32,33,34,13671735,36,37,38,39,
%U 78124840,41,42,43,44,439452945,46,47,48,49,295410156050,51
%N Number of totally ramified extensions over Q_5 with degree n in the algebraic closure of Q_5.
%D 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
%F 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)
%e 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)
%p 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;
%Y Cf. A100976, A100977, A100978, A100979, A100980, A100983, A100984, A100985, A100986.
%K nonn
%O 1,2
%A Volker Schmitt (clamsi(AT)gmx.net), Nov 25 2004
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