%I #15 May 04 2024 14:52:17
%S 1,3,10,5,2,108,2,8,795,6,2,1493,2,6,1172,13,2
%N Number of Q_3-isomorphism classes of fields of degree n in the algebraic closure of Q_3.
%D 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 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 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.
%e a(3)=10. There is the one unramified extension, three ramified cyclic extensions, six extensions with Galoisgroup S_3.
%e This gives 1+3+3*6=22 extensions (Cf. A100977) in 1+3+6=10 Q_3-isomorphism classes.
%p # for gcd(e,p)=1 only!
%p 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 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 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 p:=3; a(n):=I0Ffen(p,1,1,n);
%Y Cf. A100976, A100977, A100978, A100979, A100980, A100981, A100983, A100985.
%K nonn,hard,more
%O 1,2
%A Volker Schmitt (clamsi(AT)gmx.net), Nov 29 2004