A100983 Number of Q_2-isomorphism classes of fields of degree n in the algebraic closure of Q_2.
1, 7, 2, 59, 2, 47, 2
Offset: 1
Examples
a(4)=59: There is the one unramified extension, 8 total ramified cyclic extensions, three wildly ramified cyclic extensions, seven ( 4 total ramified, 3 tamely ramified) extensions with Galoisgroup C_2 x C_2, 36 extensions with Galoisgroup D_8 (32 total ramified, 4 wildly ramified), one extension (Q_2[x]/(x^4+2*x^3+2*x^2+2)) with Galoisgroup A_4 and, three extensions (all total ramified) with Galoisgroup S_4. This gives 1+8+3+7+2*36+4*1+4*3=107 extensions in 1+8+3+7+36+1+3=59 Q_2-isomorphism classes.
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
- Volker Schmitt, Implementation einer p-adischen Arithmetik mit darstellungstheoretischen Anwendungen, 1996
Links
- Xiang-Dong Hou and Kevin Keating, Enumeration of isomorphism classes of extensions of p-adic fields, Journal of Number Theory, Volume 104, Issue 1, January 2004, Pages 14-61.
Programs
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Maple
# for gcd(e,p)=1 only! # which means the program produces wrong values in general if n is even! 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:=2; a(n):=I0Ffen(p,1,1,n);
Formula
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.