Inverse Galois Conjecture over $\mathbb Q$

The inverse Galois problem asks whether every finite group $G$ arises as the Galois group of some finite Galois extension $L/\mathbb Q$. No single original statement of the modern problem is canonical; the conventional historical starting point is Hilbert's 1892 paper on the irreducibility of integral polynomials, whose specialization technique remains the basic tool for descending realizations from $\mathbb Q(t)$ to $\mathbb Q$ [Hilbert1892Zahlbericht].

Shafarevich proved that every finite solvable group occurs as a Galois group over $\mathbb Q$ [Shafarevich1954]. Beyond the solvable case, rigidity and related constructive methods have realized many nonsolvable families and numerous individual simple groups; the state of these techniques is surveyed in the monograph of Malle and Matzat [MalleMatzat1999]. Galois representations attached to modular forms have furnished further families of linear groups over $\mathbb Q$ [Wiese2014InverseGalois]. The regular version of the problem over $\mathbb Q(t)$ is a stronger assertion.

Despite this accumulation of realizations, no method covers all finite groups at once, and the problem remains open: a resolution requires a construction, or an abstract argument, realizing an arbitrary finite group over $\mathbb Q$.