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.

What would break it: a board pooled across every team and every method that does not separate on this question.

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. If you are working on a remaining cell, we can help with the calculation. You keep the credit.

• Take a cell. All 1,926 still empty, each 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 that refuses an impossible pair before you spend a core, an aimer, a labeller, and a checker that carries its own test. Python and PARI, no Magma seat. Same files at github.com/jonwashburn/galois.
• Ask for machines. Send a target list, a polynomial, or a construction. We will screen it, name it, or run it wide. You keep the credit.

The screen rests on seven findings, with the evidence stated beside each. Four are finite computations over the group catalog, reproducible in GAP across all 25,000 groups without building a field. Two are measured rates from the pooled Stage 1 board; those describe the recorded constructions rather than proving which fields exist.