Unsolved problems in mathematics | Galois theory

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 . 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 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. (Wikipedia).

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Related pages

Algebraic extension | Sporadic group | Order (group theory) | Galois theory | Monster group | Modular group | Algebraically closed field | Rational function | Generic polynomial | Coefficient | Hilbert's irreducibility theorem | Minimal polynomial (field theory) | Group (mathematics) | Dirichlet's theorem on arithmetic progressions | Root of unity | Function field of an algebraic variety | Polynomial | David Hilbert | Invariant theory | Group isomorphism | Alternating group | Rational number | Symmetric group | Mathieu group | Empty set | Cyclotomic field | Finite group | Simple group | Complex plane | Projective line | Characteristic (algebra) | Integer | Felix Klein | Fundamental theorem of Galois theory | Igor Shafarevich | Divisor | Cyclic group | Galois group | Eisenstein's criterion | Prime number | Irreducible polynomial | Subgroup | Automorphism group | Complex number | Parity (mathematics) | Solvable group | Galois extension | Conjugacy class | Kronecker–Weber theorem | Abelian group | Gaussian period