IGP24 / Inverse Galois research

Start with the group’s block system.

Before constructing a degree-24 field, inspect the group’s action on its 24 points. A system of twelve blocks of size two identifies the quadratic route through a degree-12 subfield. The group calculation comes first; finding the field is the arithmetic that follows.

Groups with a block system of twelve blocks of size two are realized at 98.46 percent. Groups without one are realized at 72.79 percent.

Share of groups realized, with and without the block system.

On the organizer’s 20 August 2026 board, 1,926 cells remained empty. Of these, 1,685 permit a relative-quadratic route through a degree-12 base, and 235 require a tower of another degree. The permitted bases and the limits of this group-theoretic screen are listed in the open-cell worklist. For organizer records, see the official discoveries page.

We publish tools, field data and construction targets so others can use the work. Stage 1 of the SAIR Inverse Galois Problem challenge closed on 15 August 2026.

Take a cell. Every empty cell, with the degree, group and real-place count of a base field that would write it. Deal yourself one and the page spells out the steps.
Run something. A screen, an aimer, a labeller and a checker, on Python and PARI.
Ask for machines. Send a target list, a polynomial, or a construction. We will screen it, name it, or run it wide.

The screen rests on seven findings, each labelled by how it is known.