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A100985 Number of Q_5-isomorphism classes of fields of degree n in the algebraic closure of Q_5.

%I A100985 #11 May 04 2024 14:52:44
%S A100985 1,3,2,7,26,7,2,11,3,378,2,17,2,6,1012,17,2,11,2,22302,4,6,2,29,
%T A100985 397515,6,4,14,2,406902,2,23,4,6,535732,27,2,6,4,19437446,2,15,2,14,
%U A100985 16927758,6,2,49,3
%N A100985 Number of Q_5-isomorphism classes of fields of degree n in the algebraic closure of Q_5.
%D A100985 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.
%H A100985 Xiang-Dong Hou and Kevin Keating, <a href="https://doi.org/10.1016/S0022-314X(03)00155-0">Enumeration of isomorphism classes of extensions of p-adic fields</a>, Journal of Number Theory, Volume 104, Issue 1, January 2004, Pages 14-61.
%F A100985 p:=5; 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.
%e A100985 a(3)=2. There is the one unramified extension Q_125, one ramified with Galoisgroup S_3 Q_5[x]/(x^3+5). There are 1+3*1=4 extensions (Cf. A100978) in 1+1=2 Q_5-isomorphism classes.
%p A100985 # for gcd(e,p)=1 only!
%p A100985 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:
%p A100985 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:
%p A100985 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 A100985 p:=5; a(n):=I0Ffen(p,1,1,n);
%Y A100985 Cf. A100976, A100977, A100978, A100979, A100980, A100981, A100983, A100984.
%K A100985 nonn,hard,more
%O A100985 1,2
%A A100985 Volker Schmitt (clamsi(AT)gmx.net), Nov 29 2004