Inverse Galois problem

This article describes an open problem in the following area of/related to group theory: Galois theory

This page gives information on a major conjecture
See pages on major conjectures

Statement

The inverse Galois problem is the problem of finding all Galois-realizable groups over $\mathbb {Q}$.

Associated conjecture

It is believed that the property of being Galois-realizable over $\mathbb {Q}$ is the tautology for finite groups. In other words, it is believed that every finite group can be realized as the Galois group of some Galois extension over $\mathbb {Q}$.

Thus, the inverse Galois problem is sometimes thought of as the problem of determining whether or not every finite group can be expressed as the Galois group of some Galois extension (rather than the more general problem of understanding precisely what it means for a group to be Galois-realizable).

Partial truth

Group property	Group property with finiteness	Current status	Proof/explanation
cyclic group	finite cyclic group	true	finite cyclic implies Galois-realizable; uses there are infinitely many primes that are one modulo any modulus
abelian group	finite abelian group	true	finite abelian implies Galois-realizable; uses there are infinitely many primes that are one modulo any modulus
solvable group	finite solvable group	true	Shafarevich's theorem
symmetric group	symmetric group on finite set	true	Hilbert's irreducibility theorem (?)
alternating group	alternating group on finite set	true	Hilbert's irreducibility theorem (?)

Every group of order less than $672$ is known to be Galois-realizable (see the journal article by J. Sonn. in the references section).

References

J. Sonn. "Groups of small order as Galois groups over $\mathbb {Q}$." Rocky Mountain Journal of Mathematics, 19(3) 947-956 Summer 1989. ([1])