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 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 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 let K be a field. Then the question is this: is there a Galois extension field L/K such that the Galois group of the extension is isomorphic to G? One says that G is realizable over K if such a field L exists.
There is a great deal of detailed information in particular cases. It is known that every finite group is realizable over any function field in one variable over the complex numbers C, and more generally over function fields in one variable over any algebraically closed field of characteristic zero. Shafarevich showed that every finite solvable group is realizable over Q. It is also known that every sporadic group, except possibly the Mathieu group M23, is realizable over Q.
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