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

Alex Meiburg timeroot.alex at gmail.com
Wed Nov 16 16:03:10 CET 2016


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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