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

[Charles Greathouse]

I agree, these would be great! If anyone submits them feel free to ping me
and I'll review them ASAP.

Charles Greathouse
Case Western Reserve University

On Wed, Nov 16, 2016 at 10:03 AM, Alex Meiburg <timeroot.alex at gmail.com>
wrote:

> Algebraic number fields are finite-dimensional extensions of the rationals
> that can be embedded in the complex numbers. Each has an associated number
> ring of its algebraic integers, and this has a discriminant, given by the
> square of the determinant of a matrix of its basis under different
> conjugations. (When the number ring is generated by a primitive element
> alpha, the discriminant of the ring matches with the discriminant of
> alpha's minimal polynomial). This discriminant is always an integer. 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.
>
> 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.
>
> 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 :)
>
> -- Alexander Meiburg
>
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