- Embedding problem
In
Galois theory , a branch ofmathematics , the embedding problem is a generalization of theinverse 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 calledinverse 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".
Analogously, an embedding problem for a
profinite group "F" consists of the following data: Two profinite groups "H" and "G" and two continuous epimorphisms "φ" : "F" → "G" and"f" : "H" → "G". The embedding problem is said to be finite if the group "H" is.A solution (sometimes also called weak solution) of such an embedding problem is a continuous homomorphism "γ" : "F" → "H" such that "φ" = "f" "γ". If the solution is surjective, it is called a proper solution.Properties
Finite embedding problems characterize profinite groups. The following theorem gives an illustration for this principle.
Theorem. Let "F" be a
countably (topologically) generated profinite group. Then
# "F" is projective if and only if any finite embedding problem for "F" is solvable.
# "F" is free of countable rank if and only if any finite embedding problem for "F" is properly solvable.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.
* Michael D. Fried and Moshe Jarden, "Field arithmetic", second ed., revised and enlarged by Moshe Jarden, Ergebnisse der Mathematik (3) 11, Springer-Verlag, Heidelberg, 2005.
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