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A100976 Number of all extensions over Q_2 with degree n in the algebraic closure of Q_2. 9


%S 1,7,4,107,6,124,8,6835,13,762,12,31724,14,4088,24,6999011,18,26611,

%T 20,3121122,32,98292,24,519765964,31,458738,40,267911128,30,3145704,

%U 32,1834748739523,48,9437166,48,27903655871,38,41943020,56

%N Number of all extensions over Q_2 with degree n in the algebraic closure of Q_2.

%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)=(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=2, 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(2)=7: There are 6 ramified extensions with minimal polynomials x^2+2, x^2-2, x^2+6, x^2-6, x^2+2x+2, x^2+2x+6 and one unramified x^2+x+1.

%p p:=2; 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;

%Y Cf. A100977, A100978, A100979, A100980, A100981, A100983, A100984, A100985, A100986.

%K nonn

%O 1,2

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

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Last modified December 9 22:27 EST 2019. Contains 329880 sequences. (Running on oeis4.)