In classical Galois theory for fields the inverse Galois problem is the question whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers.
This problem is ‘’inverse’‘ to the problem of computing the Galois group of some algebraic equation. It was first posed in the 19th century and is still unsolved. However there are several results dealing with special and related cases.
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Helmut Völklein, Groups as Galois Groups, an Introduction, Cambridge University Press, 1996.
‘’inverse Galois problem’‘ in wikipedia, web