Inverse Galois problem

In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers ${\displaystyle \mathbb {Q} }$. This problem, first posed in the early 19th century, is unsolved.

There are some permutation groups for which generic polynomials are known, which define all algebraic extensions of ${\displaystyle \mathbb {Q} }$ having a particular group as Galois group. These groups include all of degree no greater than 5. There also are groups known not to have generic polynomials, such as the cyclic group of order 8.

More generally, let G be a given finite group, and K a field. If there is a Galois extension field L/K whose Galois group is isomorphic to G, one says that G is realizable over K.