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A100976 Number of all extensions over Q_2 with degree n in the algebraic closure of Q_2. 9
1, 7, 4, 107, 6, 124, 8, 6835, 13, 762, 12, 31724, 14, 4088, 24, 6999011, 18, 26611, 20, 3121122, 32, 98292, 24, 519765964, 31, 458738, 40, 267911128, 30, 3145704, 32, 1834748739523, 48, 9437166, 48, 27903655871, 38, 41943020, 56 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
REFERENCES
M. Krasner, Le nombre des surcorps primitifs d'un degré donné et le nombre des surcorps métagaloisiens d'un degré donné d'un corps de nombres p-adiques. Comptes Rendus Hebdomadaires, Académie des Sciences, Paris 254, 255, 1962.
LINKS
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=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)
EXAMPLE
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.
MAPLE
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;
CROSSREFS
Sequence in context: A038270 A241359 A239602 * A152627 A277067 A354823
KEYWORD
nonn
AUTHOR
Volker Schmitt (clamsi(AT)gmx.net), Nov 24 2004
STATUS
approved

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Last modified May 20 05:07 EDT 2024. Contains 372703 sequences. (Running on oeis4.)