Inverse Galois problem

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

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 (?)