Targeted Hunter Search Calculator

Background/Instructions

The complete tables of number fields ramified at a fixed set of primes found on this server were computed by a method based on Hunter's theorem, with p-adic refinements. The same approach would let one specifically target not just primes which ramify, but also more precise local information such as the discriminant for the local p-adic algebra. The calculator below estimates the minimum number of polynomials a computer search would have to examine for a search based on Hunter's theorem. The user selects the degree, and then a combination of "local targets".

• A local target at infinity is the same as the signature of the field. Here we use half the number of complex places (traditionally called r₂).
• At a finite prime, we allow a local target to be the prime's exponent in the discriminant, or a partition of the degree n. In the latter case, consider the local algebra after tensoring with Q_p^ur, the unramified closure of Q_p. Decompose this Q_p^ur-algebra as a product of fields. The degrees of these fields over Q_p^ur is the partition of the degree.

  Said another way, the partition is the ramification degrees e_i counted with multiplicity according to the residue field degrees. So a sextic field with e=2 and f=3 counts as the partition of 6=2+2+2.

The user can select any combination of local targets, and the estimated number of polynomials is updated. It is labeled "Volume" since one can think of this as the volume of a region to be searched.

The difficulty rating is simply the log base 10 of the volume. Each local target is labeled by either a partition (with "ones" omitted) or by a discriminant exponent, along with the difficulty rating for that particular target. Each column which has some items selected will also display its contribution to the difficulty rating at the bottom of its column. The total difficulty is the sum of the local difficulties. (Note, the difficulties of targets in a given column do not add to give the total difficulty for the column.)

There is one button whose function may not be immediately clear. It is initially marked Partn, and can also be marked as Discs. This button toggles the choice of local targets from Discriminant exponents to Partitions (when it is marked Discs), and back to Partitions (when it is marked Partn).

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