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Third SAIR competition: inverse Galois challenge

16 June, 2026 in advertising, math.GR, math.NT | Tags: Andrew Sutherland, competition, David Roe, Galois theory, Jen Paulhus, John Jones, SAIR | by Terence Tao

I am happy to announce the third SAIR challenge, which is focused on obtaining numerical data for the infamous inverse Galois problem. This is a collaborative project with the L-functions and modular forms database (LMFDB), and is organized by John Jones, Jen Paulhus, David Roe, Andrew Sutherland, and myself. The challenge is somewhat similar to my own Equational Theories Project, in that one is trying to complete a large mathematical data set in a verified fashion, except that the target data set had an existing mathematical interest. Also, the verification will be done by MAGMA (as well as PARI/GP) rather than Lean.

Let me first quickly review the inverse Galois problem. Suppose one has an irreducible polynomial of one variable of some degree and integer coefficients; take for instance . Then will have distinct roots ; in this case the roots happen to be

The roots generate some degree extension of the rational numbers . Any automorphism of this field extension must permute the roots , and thus generates a subgroup of the permutation group (defined up to relabeling of the roots), which we call the Galois group of . This is some subgroup of that acts transitively on the roots (because each root generates the field). Typically, it is all of ; but occasionally it is smaller. For example, the particular cubic polynomial above has the special property that each root individually generates the entire field , thanks to the identities

Because of this, the Galois group of is the cyclic group (or equivalently, the alternating group ), rather than the full symmetric group . (This is in contrast to, say, , whose roots , , cannot be expressed as rational polynomials of each other, and whose Galois group is all of .) In fact, in the cubic case, it turns out that the Galois group is when the discriminant is a perfect square, and otherwise.

More generally, we have

Problem 1 (Inverse Galois Problem) Let be a transitive permutation group on letters. Can be realized as the Galois group of some degree irreducible polynomial with integer coefficients (after identifying the roots of suitably with the letters)?

The answer to this problem is known to be positive for , with the single possible exception of the sporadic Mathieu group : there are transitive permutation groups on letters (cf. OEIS A002106), and for of them, a polynomial has been located with that Galois group; see this database of Klüners and Malle. The problem of locating a polynomial with Galois group is a notorious open problem, though this is likely to be quite a difficult problem, and not the objective of the SAIR challenge.

Instead, we will focus on "breadth" rather than "depth", in order to leverage the power of crowdsourcing and modern AI technologies. It turns out that there are distinct transitive permutation groups on letters, which are conventionally labeled from (the cyclic group ) to (the permutation group ). The first stage of the challenge will be:

Problem 2 (First stage of SAIR challenge) For as many of the groups , , locate an integer polynomial with that Galois group (up to isomorphism). (Also of interest is to specify the number of real roots, and to keep the discriminant low; more on this later.)

The verification side of this problem is essentially solved: the MAGMA computer algebra system can take any candidate polynomial and locate its Galois group within seconds. The MAGMA team has kindly granted SAIR a limited license to provide an API for contestants to calculate a certain number of Galois groups per day without needing to purchase their own license, though of course they are free to use their other computational tools to also perform these calculations outside of the competition.

The LMFDB already has polynomials for 286 of the 25000 groups, so there is plenty of remaining polynomials to claim in the challenge.

For applications, it is of interest to track some other statistics of a polynomial besides its Galois group. One of these is the number of real roots, which is a number between and of the same parity as (and which has to be achievable as the number of fixed points of one of the permutations in the Galois group, namely the one corresponding to complex conjugation); in particular, this number must be even in the degree case. Combining the label of the Galois group with the number of roots turns out to generate pairs in degree , and the challenge is actually to attach polynomials to as many of these pairs as possible. (The LMFDB has...