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A100980 Number of totally ramified extensions over Q_3 with degree n in the algebraic closure of Q_3. 8


%S 1,2,21,4,5,150,7,8,5085,10,11,2892,13,14,10905,16,17,984114,19,20,

%T 137739,22,23,472344,25,26,900792441,28,29,5314350,31,32,17537487,34,

%U 35,13832346276,37,38,186535713,40,41,602654010,43,44,1408273477425

%N Number of totally ramified extensions over Q_3 with degree n in the algebraic closure of Q_3.

%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=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)

%e 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)

%p 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;

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

%K nonn

%O 1,2

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

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Last modified December 5 22:37 EST 2019. Contains 329782 sequences. (Running on oeis4.)