Inverse Galois problem
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In mathematics, the inverse Galois problem concerns whether or not we can find a rational field extension with a given Galois group. To be a little more precise, let G be a given finite group, and let Q be the field of rational numbers. Then the question is this: is there a Galois extension field L/Q such that the Galois group of the extension is isomorphic with G? One says that G is realizable over Q if such a field L exists.
Hilbert showed that this question is related to a rationality question for G: if K is any extension of the rational field Q, on which G acts as an automorphism group and the invariant field KG is rational over Q, then G is realizable over Q. Here rational means that it is a pure transcendental extension of Q, generated by an algebraically independent set. This criterion can for example be used to show that all the symmetric groups are realizable.
Much detailed work has been carried out on the question, which is in no sense solved in general. Some of this is based on constructing G geometrically as a Galois covering of the projective line: in algebraic terms, start with an extension of the field Q(t) of rational functions in an indeterminate t. After that, one applies Hilbert's irreducibility theorem to specialise t, in such a way as to preserve the Galois group.
