Polynomials with Galois group 17T7

Noam Elkies (Harvard)

The inverse Galois problem asks: Given a finite group $G$, is there a Galois extension $F/{\bf Q}$ with ${\rm Gal}(F/{\bf Q}) \cong G$ ? The problem is still unsolved, but many special cases are known, including all sporadic groups except the Mathieu group $M_{23}$ , and all transitive subgroups of $S_n$ for $n < 23$ . The latter result is quite recent: $S_{17}$ has a transitive subgroup “17T7” for which the existence of such $F$ has been known for only a year or so and explicit examples were first exhibited in early 2024.

The group 17T7 is the extension of ${SL}2({\bf F}_{16})$ by the Galois involution of the extension of finite fields ${\bf F}_{16} / {\bf F}_4$ , acting on the 17 points of the projective line over ${\bf F}_{16}$ . After introducing the inverse Galois problem, we explain why 17T7 was so difficult (whereas examples of the related groups 17T6 and 17T8 had been known for years), and outline the theoretical and computational work that led to examples such as the splitting field of

$x^{17} - 2 x^{16} + 12 x^{15} - 28 x^{14} + 60 x^{13} - 160 x^{12}+200 x^{11} - 500 x^{10} + 705 x^9 - 886 x^8 + 2024 x^7 - 604 x^6+2146 x^5 + 80 x^4 - 1376 x^3 - 496 x^2 - 1013 x - 490.$