inverse Galois problem

Galois theory hands you a group from every polynomial. The inverse Galois problem turns the arrow around and asks: starting from an abstract finite group, can you always find a polynomial over the rationals whose symmetries are exactly that group? Surprisingly, despite a century and a half of effort, nobody knows whether the answer is always yes.

Stated precisely, the inverse Galois problem asks whether every finite group G occurs as the Galois group Gal(L/Q) of some finite Galois extension L of the rational numbers. The problem is wide open in general. It is known to have an affirmative answer for many classes: all solvable groups (Shafarevich's theorem), all symmetric and alternating groups, and 25 of the 26 sporadic simple groups (the Mathieu group M_23 being the famous holdout at the time of writing).

A major tool is rigidity and the theory of Hurwitz spaces: realizing G over the function field Q(t) and then specializing t, often via Hilbert's irreducibility theorem, to descend to Q. One must be careful — the analogous problem over other base fields can behave very differently; for instance every finite group is a Galois group over the field C(t) of rational functions, so the difficulty is genuinely about the arithmetic of Q, not just group theory.