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A100984
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Number of Q_3-isomorphism classes of fields of degree n in the algebraic closure of Q_3.
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8
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1, 3, 10, 5, 2, 108, 2, 8, 795, 6, 2, 1493, 2, 6, 1172, 13, 2
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OFFSET
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1,2
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REFERENCES
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Xiang-Dong Hou and Kevin Keating, Enumeration of isomorphism classes of extensions of p-adic fields, 2001
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
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LINKS
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FORMULA
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p:=3; n=f*e; f residue degree, e ramification index if (p, e)=1, let I(f, e):=b/e*sum_{h=0}^{e-1} 1/c_h, where b=gcd(e, p^f-1), c_h the smallest positive integer such that b divides (p^c-1)*h a(n) = sum_{f | n} I(f, n/f) There exists a formula, when p divides e exactly and there exists a big formula for some cases when p^2 divides e exactly.
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EXAMPLE
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a(3)=10, There is the one unramified extension, three ramified cyclic extensions, six extensions with Galoisgroup S_3
This gives 1+3+3*6=22 extensions (Cf. A100977) in 1+3+6=10 Q_3-isomorphism classes.
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MAPLE
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# for gcd(e, p)=1 only! smallestIntDiv:=proc() local b, q, h, i; b:=args[1]; q:=args[2]; h:=args[3]; for i from 1 to infinity do if gcd(b, (q^i-1)*h)=b then return i; fi; od; end: I0Ffefe:=proc() local p, f1, e1, f, e, i, q, h, summe, c, b; p:=args[1]; f1:=args[2]; e1:=args[3]; f:=args[4]; e:=args[5]; summe:=0; q:=p^f1; b:=gcd(e, q^f-1); for h from 0 to e-1 do c:=smallestIntDiv(b, q, h); summe:=summe+1/c; od; return b/e*summe; end: I0Ffen:=proc() local p, e1, f1, n, f, e, summe; p:=args[1]; e1:=args[2]; f1:=args[3]; n:=args[4]; summe:=0; for f in divisors(n) do e:=n/f; summe:=summe+I0Ffefe(p, f1, e1, f, e); od; return summe; end: p:=3; a(n):=I0Ffen(p, 1, 1, n);
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CROSSREFS
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KEYWORD
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hard,nonn
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AUTHOR
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Volker Schmitt (clamsi(AT)gmx.net), Nov 29 2004
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STATUS
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approved
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