[seqfan] Re: Sequence request: no. of algebraic number fields with a given discriminant

[Chris Thompson]

On Nov 16 2016, Alex Meiburg wrote:

[Description of algebraic number fields snipped]
>                                                                 It is
>a well-known theorem that there are finitely many fields with a given
>discriminant... it seems natural to ask how many fields there are with a
>given discriminant.

Yes, but ...

>This wouldn't be very convenient to compute in automated fashion, but
>literature should contain values for many values. This should be stored as
>all of:
>- Number of fields with discriminant n
>- Number of fields with discriminant -n
>- The previous two, interleaved -- so disc=1, -1, 2, -2, 3, -3 etc.
>- Number of fields with |discriminant| = n.

Note that (a) discriminants are always 0 or 1 mod 4. and (b) your
proposed numbers would be rather dominated by the (boring) discriminants
for quadratic fields, at least for small values of the discriminant.

>This doesn't appear to be OEIS, but I was just searching by keyword since I
>don't have these values myself. References + submission would probably be a
>good thing, thing seems like a reasonably fundamental thing to count :)

What we *do* have in OEIS is various lists of discriminants for number
fields with a given degree and signature:

  A006832  totally real cubic fields [signature (3,0)]
  A023679  complex cubic fields [signature (1,1)]
  A023680  totally real quartic fields [signature (4,0)]
  A023681  quartic fields with signature (2,1)
  A023682  totally complex quartic fields [signature (0,2)]
  A023683  totally real quintic fields [signature (5,0)]
  A023684  quintic fields with signature (3,1)
  A023685  quintic fields with signature (1,2)
  A023686  totally real sextic fields [signature (6,0)]
  A023687  totally complex sextic fields [signature (0,3)]

For completeness: here an algebraic number field of signature (r,s)
is one that has r real embeddings and s pairs of non-real complex
embeddings, its degree thus being r+2s. Separating the fields by
signature at least solves the problem of how to deal with the sign
of the discriminant, as this is always (-1)^s.

The above list is certainly due to taking the initial data from
the 1989 book by Pohst and Zassenhaus. Indeed, that gives only
the least discriminant for the sextic fields with signatures
(4,1) and (2,2), and I suspect Neil balked at putting in
sequences with only one number :-)

[Incidentally, I was glad to see that the one case of non-isomorphic
number fields with the same signature and discriminant covered by
the tables in Pohst & Zassenhaus is correctly represented in
A023682 by 576 occurring twice!]

The only ones of these that have been [mildly] extended since
then seem to be the totally real cases. There is a lot of data
out there by now to substantially extend both the individual
sequences and the range of signatures covered. For example, we
could use the material at

  http://pari.math.u-bordeaux.fr/pub/pari/packages/nftables/

-- 
Chris Thompson
Email: cet1 at cam.ac.uk