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''?
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== * {{cite book |mr=260875 | zbl=0221.12013 | title=Introduction of profinite groups and Galois cohomology | series=Queen's Pap. Pure Appl. Math. | year=1970 | volume=24 | publisher=Queen's University, Kingston, Ontario }} * {{cite book |doi=10.1090/mmono/165 |title=The Embedding Problem in Galois Theory |series=Translations of Mathematical Monographs |year=1997 |volume=165 |isbn=9780821845929 }} * {{cite book |last1=Fried |first1=Michael D. |last2=Jarden |first2=Moshe |author-link=Michael D. Fried |author-link2=Moshe Jarden|doi=10.1007/978-3-540-77270-5 |title=Field Arithmetic |series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics |year=2008 |volume=11 |isbn=978-3-540-77269-9 }} * {{cite book |doi=10.1090/fim/021 |title=Brauer Type Embedding Problems |series=Fields Institute Monographs |year=2005 |volume=21 |isbn=9780821837269 }} * Vahid Shirbisheh, ''Galois embedding problems with abelian kernels of exponent p'' VDM Verlag Dr. Müller, {{isbn|978-3-639-14067-5}}, (2009). Category:Group theory Category:Galois theory