Inverse Galois problem
This article describes an open problem in the following area of/related to group theory: Galois theory
The inverse Galois problem is the problem of finding all Galois-realizable groups over .
It is believed that the property of being Galois-realizable over 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 .
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).
|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 (?)|