Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...

, a branch of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the embedding problem is a generalization of the

inverse Galois problem In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers \mathbb. This problem, first posed in the early 19th century, is unsolved. There ...

. Roughly speaking, it asks whether a given

Galois extension In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ...

can be embedded into a Galois extension in such a way that the

restriction map A restriction map is a map of known restriction sites within a sequence of DNA. Restriction mapping requires the use of restriction enzymes. In molecular biology, restriction maps are used as a reference to engineer plasmids or other relatively ...

between the corresponding

Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...

s is given.

Definition

Given a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

''K'' and a finite

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

''H'', one may pose the following question (the so called

). 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 In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. ...

''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 In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...

(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. * V. V. Ishkhanov, B. B. Lur'e, D. K. Faddeev, ''The embedding problem in Galois theory'' Translations of Mathematical Monographs, vol. 165, American Mathematical Society (1997). * 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. * A. Ledet, ''Brauer type embedding problems'' Fields Institute Monographs, no. 21, (2005). * Vahid Shirbisheh, ''Galois embedding problems with abelian kernels of exponent p''

VDM Verlag Dr. Müller Omniscriptum Publishing Group, formerly known as VDM Verlag Dr. Müller, is a German publishing group headquartered in Riga, Latvia. Founded in 2002 in Düsseldorf, its book production is based on print-to-order technology. The company publis ...

, {{isbn, 978-3-639-14067-5, (2009). * Almobaideen Wesam, Qatawneh Mohammad, Sleit Azzam, Salah Imad,
Efficient mapping scheme of ring topology onto tree-hypercubes
''
Journal of Applied Sciences
2007 Group theory Galois theory