Embedding problem

In Galois Theory, a branch of mathematics, the embedding problem is a generalization of the inverse Galois problem. Roughly speaking, it asks whether a given Galois extension can be embedded into a Galois extension in such a way that the restriction map between the corresponding Galois groups is given.

Definition

Given a field K and a finite group H, one may pose the following question (the so called inverse Galois problem). Is there a Galois extension F/K with Galois group isomorphic to H. The embedding problem is a generalization of this problem:

Let L/K be a Galois extension with Galois group G and let f : H → G be an epimorphism. Is there a Galois extension F/K with Galois group H and an embedding α : L → F fixing K under which the restriction map from the Galois group of F/K to the Galois group of L/K coincides with f.

References

• Luis Ribes, Introduction to Profinite groups and Galois cohomology (1970), Queen's Papers in Pure and Appl. Math., no. 24, Queen's university, Kingstone, Ont.