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.
Volker Schmitt, Implementation einer p-adischen Arithmetik mit darstellungstheoretischen Anwendungen, 1996
LINKS
Robin Visser, Table of n, a(n) for n = 1..3000
Chad Awtrey, Brett Barkley, Nicole E. Miles, Christopher Shill, and Erin Strosnider, Degree 12 2-adic fields with automorphism group of order 4, Rocky Mountain J. Math. 45 (2015), no. 6, 1755-1764.
Chad Awtrey, Nicole Miles, Jonathan Milstead, Christopher Shill, and Erin Strosnider, Degree 14 2-adic fields, Involve 8 (2015), no. 2, 329-336.
Jordi Guàrdia-Rúbies, John W. Jones, Kevin Keating, Sebastian Pauli, David P. Roberts, and David Roe, Families of p-adic fields, arXiv:2507.02360 [math.NT], 2025.
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.
John W. Jones and David P. Roberts, A database of local fields, J. Symbolic Comput. 41 (2006), no. 1, 80-97.
John W. Jones and David P. Roberts, Octic 2-adic fields, J. Number Theory 128 (2008), no. 6, 1410-1429.
LMFDB, Local field count results.
Maurizio Monge, Determination of the number of isomorphism classes of extensions of a p-adic field, J. Number Theory 131 (2011), no. 8, 1429-1434.
Sebastian Pauli and Xavier-François Roblot, On the computation of all extensions of a p-adic field of a given degree, Math. Comp. 70 (2001), no. 236, 1641-1659.
FORMULA
p:=2; n=f*e; f residue degree, e ramification index. If gcd(p, e)=1, let I(f, e):=b/e*Sum_{h=0..e-1} 1/c_h, where b=gcd(e, p^f-1), and c_h is the smallest positive integer such that b divides (p^c_h-1)*h. Then a(n) = Sum_{f | n} I(f, n/f) [Hou-Keating, Proposition 4.1]. There exists a formula when p divides e exactly (see Hou-Keating, Theorem 5.6) and there exists a big formula for some cases when p^2 divides e exactly (see Hou-Keating, Theorem 10.1).
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 Galois group C_2 x C_2, 36 extensions with Galois group D_8 (32 total ramified, 4 wildly ramified), one extension (Q_2[x]/(x^4+2*x^3+2*x^2+2)) with Galois group A_4, and three extensions (all total ramified) with Galois group 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!
# 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);
CROSSREFS
KEYWORD
nonn
AUTHOR
Volker Schmitt (clamsi(AT)gmx.net), Nov 29 2004
EXTENSIONS
More terms from Robin Visser, Nov 29 2025
STATUS
approved