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Über das Umkehrproblem der Galoisschen Theorie. (On the inverse problem of Galois theory). (German) Zbl 0662.12008
Let K be a field, f(x)\(\in K[x]\) a polynomial. Then to f(x) is associated a group: if E is a splitting field for f(x) over K then Gal(E/K) is the group associated to f(x) called the Galois group of f(x) over K. The inverse problem of Galois theory for K is: Given a finite group G does there exist a polynomial f(x)\(\in K[x]\) such that the Galois group of f(x) over K is isomorphic to G? - The problem for \(K={\mathbb{Q}}\) is still open inspite of contributions by giants for 150 years.
The author gives an excellent survey of the status of the problem.
Reviewer: T.Soundararajan

11R32 Galois theory
12-03 History of field theory
12F10 Separable extensions, Galois theory