|
| |
|
|
A100985
|
|
Number of Q_5-isomorphism classes of fields of degree n in the algebraic closure of Q_5.
|
|
8
|
|
|
|
1, 3, 2, 7, 26, 7, 2, 11, 3, 378, 2, 17, 2, 6, 1012, 17, 2, 11, 2, 22302, 4, 6, 2, 29, 397515, 6, 4, 14, 2, 406902, 2, 23, 4, 6, 535732, 27, 2, 6, 4, 19437446, 2, 15, 2, 14, 16927758, 6, 2, 49, 3
(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
|
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.
|
|
|
EXAMPLE
|
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.
|
|
|
MAPLE
|
# 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:=5; a(n):=I0Ffen(p, 1, 1, n);
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,hard,more
|
|
|
AUTHOR
|
Volker Schmitt (clamsi(AT)gmx.net), Nov 29 2004
|
|
|
STATUS
|
approved
|
| |
|
|
