cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A100976 Number of all extensions over Q_2 with degree n in the algebraic closure of Q_2.

Original entry on oeis.org

1, 7, 4, 107, 6, 124, 8, 6835, 13, 762, 12, 31724, 14, 4088, 24, 6999011, 18, 26611, 20, 3121122, 32, 98292, 24, 519765964, 31, 458738, 40, 267911128, 30, 3145704, 32, 1834748739523, 48, 9437166, 48, 27903655871, 38, 41943020, 56
Offset: 1

Views

Author

Volker Schmitt (clamsi(AT)gmx.net), Nov 24 2004

Keywords

Examples

			a(2)=7: There are 6 ramified extensions with minimal polynomials x^2+2, x^2-2, x^2+6, x^2-6, x^2+2x+2, x^2+2x+6 and one unramified x^2+x+1.

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.

Crossrefs

Cf. A100977, A100978, A100979, A100980, A100981, A100983, A100984, A100985, A100986.

Programs

  • Maple
    p:=2; eps:=proc()local p,s,i,sum; p:=args[1]; s:=args[2]; if s=-1 then return -infinity; fi; if s=0 then return 0; fi; sum:=0; for i from 1 to s do sum:=sum+1/p^i; od; return sum; end: ppart:=proc() local p,n; p:=args[1]; n:=args[2]; return igcd(n,p^n); end: qpart:=proc() local p,n; p:=args[1]; n:=args[2]; return n/igcd(n,p^n); end: logp:=proc() local p, pp; p:=args[1]; pp:=args[2]; if op(ifactors(pp))[2]=[] then return 0; else return op(op(ifactors(pp))[2])[2]; fi; end: summe:=0; m:=logp(p, ppart(p,n)); h:=qpart(p,n); for s from 0 to m do summe:=summe+(p^(m+s+1)-p^(2*s))/(p-1)*(p^(eps(p,s)*n)-p^(eps(p,s-1)*n)); od; a(n):=sigma(h)*summe;

Formula

a(n)=(sum_{d|h}d)*(sum_{s=0}^m (p^(m+s+1)-p^(2*s))/(p-1)*(p^(eps(s)*n)-p^(eps(s-1)*n))), where p=2, n=h*p^m, with gcd(h, p)=1, eps(-1)=-infinity, eps(0)=0 and eps(s)=sum_{i=1 to s} 1/(p^i)

A100977 Number of all extensions over Q_3 with degree n in the algebraic closure of Q_3.

Original entry on oeis.org

1, 3, 22, 7, 6, 228, 8, 15, 5323, 18, 12, 5068, 14, 24, 13092, 31, 18, 1495839, 20, 42, 157424, 36, 24, 885660, 31, 42, 942953404, 56, 30, 9565848, 32, 63, 19131816, 54, 48, 24240086731, 38, 60, 200884628, 90, 42, 1033121184, 44, 84
Offset: 1

Views

Author

Volker Schmitt (clamsi(AT)gmx.net), Nov 24 2004

Keywords

Examples

			a(2)=3 There are 2 ramified extensions with minimal polynomials x^2+3, x^2-3 and one unramified x^2+2*x+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.

Crossrefs

Cf. A100976, A100978, A100979, A100980, A100981, A100983, A100984, A100985, A100986.

Programs

  • Maple
    p:=3; eps:=proc()local p,s,i,sum; p:=args[1]; s:=args[2]; if s=-1 then return -infinity; fi; if s=0 then return 0; fi; sum:=0; for i from 1 to s do sum:=sum+1/p^i; od; return sum; end: ppart:=proc() local p,n; p:=args[1]; n:=args[2]; return igcd(n,p^n); end: qpart:=proc() local p,n; p:=args[1]; n:=args[2]; return n/igcd(n,p^n); end: logp:=proc() local p, pp; p:=args[1]; pp:=args[2]; if op(ifactors(pp))[2]=[] then return 0; else return op(op(ifactors(pp))[2])[2]; fi; end: summe:=0; m:=logp(p, ppart(p,n)); h:=qpart(p,n); for s from 0 to m do summe:=summe+(p^(m+s+1)-p^(2*s))/(p-1)*(p^(eps(p,s)*n)-p^(eps(p,s-1)*n)); od; a(n):=sigma(h)*summe;

Formula

a(n)=(sum_{d|h}d)*(sum_{s=0}^m (p^(m+s+1)-p^(2*s))/(p-1)*(p^(eps(s)*n)-p^(eps(s-1)*n))), where p=3, n=h*p^m, with gcd(h, p)=1, eps(-1)=-infinity, eps(0)=0 and eps(s)=sum_{i=1 to s} 1/(p^i)

A100978 Number of all extensions over Q_5 with degree n in the algebraic closure of Q_5.

Original entry on oeis.org

1, 3, 4, 7, 106, 12, 8, 15, 13, 1818, 12, 28, 14, 24, 12424, 31, 18, 39, 20, 109242, 32, 36, 24, 60, 8281131, 42, 40, 56, 30, 4687272, 32, 63, 48, 54, 15624848, 91, 38, 60, 56, 146484090, 42, 96, 44, 84, 634765378, 72, 48, 124, 57
Offset: 1

Views

Author

Volker Schmitt (clamsi(AT)gmx.net), Nov 24 2004

Keywords

Examples

			a(2)=3 There are 2 ramified extensions with minimal polynomials x^2-5, x^2-10 and one unramified x^2+4*x+2.

References

  • 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

Crossrefs

Cf. A100976, A100977, A100979, A100980, A100981, A100983, A100984, A100985, A100986.

Programs

  • Maple
    p:=5; eps:=proc()local p,s,i,sum; p:=args[1]; s:=args[2]; if s=-1 then return -infinity; fi; if s=0 then return 0; fi; sum:=0; for i from 1 to s do sum:=sum+1/p^i; od; return sum; end: ppart:=proc() local p,n; p:=args[1]; n:=args[2]; return igcd(n,p^n); end: qpart:=proc() local p,n; p:=args[1]; n:=args[2]; return n/igcd(n,p^n); end: logp:=proc() local p, pp; p:=args[1]; pp:=args[2]; if op(ifactors(pp))[2]=[] then return 0; else return op(op(ifactors(pp))[2])[2]; fi; end: summe:=0; m:=logp(p, ppart(p,n)); h:=qpart(p,n); for s from 0 to m do summe:=summe+(p^(m+s+1)-p^(2*s))/(p-1)*(p^(eps(p,s)*n)-p^(eps(p,s-1)*n)); od; a(n):=sigma(h)*summe;

Formula

a(n)=(sum_{d|h}d)*(sum_{s=0}^m (p^(m+s+1)-p^(2*s))/(p-1)*(p^(eps(s)*n)-p^(eps(s-1)*n))), where p=5, n=h*p^m, with gcd(h, p)=1, eps(-1)=-infinity, eps(0)=0 and eps(s)=sum_{i=1 to s} 1/(p^i)

A100979 Number of totally ramified extensions over Q_2 with degree n in the algebraic closure of Q_2.

Original entry on oeis.org

1, 6, 3, 92, 5, 90, 7, 5880, 9, 630, 11, 23028, 13, 3570, 15, 6021104, 17, 18414, 19, 2580460, 21, 90090, 23, 377290728, 25, 425958, 27, 233963492, 29, 1966050, 31, 1578396286944, 33, 8912862, 35, 19308478428, 37, 39845850, 39, 108
Offset: 1

Views

Author

Volker Schmitt (clamsi(AT)gmx.net), Nov 25 2004

Keywords

Examples

			a(2)=6 There are 6 ramified extensions with minimal polynomials x^2+2, x^2-2, x^2+6, x^2-6, x^2+2x+2, x^2+2x+6, there is another one by x^2+x+1, but this is unramified.

References

  • 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

Crossrefs

Cf. A100976, A100977, A100978, A100980, A100981, A100983, A100984, A100985, A100986.

Programs

  • Maple
    p:=2; eps:=proc()local p,s,i,sum; p:=args[1]; s:=args[2]; if s=-1 then return -infinity; fi; if s=0 then return 0; fi; sum:=0; for i from 1 to s do sum:=sum+1/p^i; od; return sum; end: ppart:=proc() local p,n; p:=args[1]; n:=args[2]; return igcd(n,p^n); end: qpart:=proc() local p,n; p:=args[1]; n:=args[2]; return n/igcd(n,p^n); end: logp:=proc() local p, pp; p:=args[1]; pp:=args[2]; if op(ifactors(pp))[2]=[] then return 0; else return op(op(ifactors(pp))[2])[2]; fi; end: summe:=0; m:=logp(p, ppart(p,n)); h:=qpart(p,n); for s from 0 to m do summe:=summe+(p^s*(p^(eps(p,s)*n)-p^(eps(p,s-1)*n)); od; a(n):=n*summe;

Formula

a(n)=n*(sum_{s=0}^m p^s*(p^(eps(s)*n)-p^(eps(s-1)*n))), where p=2, n=h*p^m, with gcd(h, p)=1, eps(-1)=-infinity, eps(0)=0 and eps(s)=sum_{i=1 to s} 1/(p^i)

A100981 Number of totally ramified extensions over Q_5 with degree n in the algebraic closure of Q_5.

Original entry on oeis.org

1, 2, 3, 4, 105, 6, 7, 8, 9, 1210, 11, 12, 13, 14, 9315, 16, 17, 18, 19, 62420, 21, 22, 23, 24, 8203025, 26, 27, 28, 29, 2343630, 31, 32, 33, 34, 13671735, 36, 37, 38, 39, 78124840, 41, 42, 43, 44, 439452945, 46, 47, 48, 49, 295410156050, 51
Offset: 1

Views

Author

Volker Schmitt (clamsi(AT)gmx.net), Nov 25 2004

Keywords

Examples

			a(3)=3: there is one totally ramified extension with Galois group S_3, so there are 3 totally ramified extensions in the algebraic closure all isomorphic to Q_5[x]/(x^3+5)

References

  • 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

Crossrefs

Cf. A100976, A100977, A100978, A100979, A100980, A100983, A100984, A100985, A100986.

Programs

  • Maple
    p:=5; eps:=proc()local p,s,i,sum; p:=args[1]; s:=args[2]; if s=-1 then return -infinity; fi; if s=0 then return 0; fi; sum:=0; for i from 1 to s do sum:=sum+1/p^i; od; return sum; end: ppart:=proc() local p,n; p:=args[1]; n:=args[2]; return igcd(n,p^n); end: qpart:=proc() local p,n; p:=args[1]; n:=args[2]; return n/igcd(n,p^n); end: logp:=proc() local p, pp; p:=args[1]; pp:=args[2]; if op(ifactors(pp))[2]=[] then return 0; else return op(op(ifactors(pp))[2])[2]; fi; end: summe:=0; m:=logp(p, ppart(p,n)); h:=qpart(p,n); for s from 0 to m do summe:=summe+(p^s*(p^(eps(p,s)*n)-p^(eps(p,s-1)*n)); od; a(n):=n*summe;

Formula

a(n)=n*(sum_{s=0}^m p^s*(p^(eps(s)*n)-p^(eps(s-1)*n))), where p=5, n=h*p^m, with gcd(h, p)=1, eps(-1)=-infinity, eps(0)=0 and eps(s)=sum_{i=1 to s} 1/(p^i)

A100983 Number of Q_2-isomorphism classes of fields of degree n in the algebraic closure of Q_2.

Original entry on oeis.org

1, 7, 2, 59, 2, 47, 2
Offset: 1

Views

Author

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

Keywords

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

Crossrefs

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

Programs

  • 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.

A100984 Number of Q_3-isomorphism classes of fields of degree n in the algebraic closure of Q_3.

Original entry on oeis.org

1, 3, 10, 5, 2, 108, 2, 8, 795, 6, 2, 1493, 2, 6, 1172, 13, 2
Offset: 1

Views

Author

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

Keywords

Examples

			a(3)=10. There is the one unramified extension, three ramified cyclic extensions, six extensions with Galoisgroup S_3.
This gives 1+3+3*6=22 extensions (Cf. A100977) in 1+3+6=10 Q_3-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.

Links

Crossrefs

Cf. A100976, A100977, A100978, A100979, A100980, A100981, A100983, A100985.

Programs

  • 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:=3; a(n):=I0Ffen(p,1,1,n);

Formula

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.

A100985 Number of Q_5-isomorphism classes of fields of degree n in the algebraic closure of Q_5.

Original entry on oeis.org

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
Offset: 1

Views

Author

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

Keywords

Examples

			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.

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

Crossrefs

Cf. A100976, A100977, A100978, A100979, A100980, A100981, A100983, A100984.

Programs

  • 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);

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.
Showing 1-8 of 8 results.

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