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Inverse Galois problem
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Unsolved problem in mathematics:
Is every finite group the Galois group of a Galois extension of the rational numbers?
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,[1] 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.
Contents
1 Partial results
2 A simple example: cyclic groups
2.1 Worked example: the cyclic group of order three
3 Symmetric and alternating groups
3.1 Alternating groups
3.1.1 Odd Degree
3.1.2 Even Degree
4 Rigid groups
5 A construction with an elliptic modular function
6 Notes
7 References
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