## 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

### MSC:

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

### Keywords:

inverse problem of Galois theory