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A100983 Number of Q_2-isomorphism classes of fields of degree n in the algebraic closure of Q_2. 8
1, 7, 2, 59, 2, 47, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

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

Volker Schmitt, Implementation einer p-adischen Arithmetik mit darstellungstheoretischen Anwendungen, 1996

LINKS

Table of n, a(n) for n=1..7.

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.

EXAMPLE

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.

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:=2; a(n):=I0Ffen(p, 1, 1, n);

CROSSREFS

Cf. A100976, A100977, A100978, A100979, A100980, A100981, A100984, A100985, A100986.

Sequence in context: A248189 A096040 A038268 * A103237 A239120 A021582

Adjacent sequences:  A100980 A100981 A100982 * A100984 A100985 A100986

KEYWORD

hard,nonn

AUTHOR

Volker Schmitt (clamsi(AT)gmx.net), Nov 29 2004

STATUS

approved

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Last modified November 22 08:33 EST 2019. Contains 329389 sequences. (Running on oeis4.)