The inverse Galois problem asks whether every finite group 
is isomorphic to the Galois
group 
of some finite Galois extension field 
. More generally, for a field 
, a 
-extension of 
is a finite Galois extension
field 
whose Galois group 
is isomorphic
to 
.
A stronger form asks for a regular Galois extension of 
whose Galois group is isomorphic
to 
.
If 
,
Hilbert's irreducibility theorem
implies that specializing 
to suitable values in 
gives infinitely many 
-extensions of 
.
Constructions published between 1984 and 1989 purported to realize 25 of the 26 sporadic groups as Galois
groups over 
.
There is a caveat to this usual summary. The original construction for the baby
monster group was later found to contain a calculation error, although a correct
realization was subsequently obtained. The Mathieu
group M23 remained the sole exception until Huang et al.
(2026) constructed a regular Galois extension
of 
with Galois group 
and an explicit degree-23 polynomial
over 
whose splitting field has Galois
group 
.
Consequently, for every sporadic group 
and every number field 
, there is a regular 
-extension of 
and a 
-extension of 
. There is also an infinite mutually independent collection
of 
-extensions
of 
.
Here mutually independent means that, for every 
, the smallest field containing
any 
distinct members has Galois group isomorphic
to the 
-fold
direct product 
over 
.
Is a Galois Group over 
." Aug. 8, 2026.