In Galois theory

Galois theory

In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...

, a branch of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the embedding problem is a generalization of the inverse Galois problem

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 Q. This problem, first posed in the 19th century, is unsolved....

. Roughly speaking, it asks whether a given Galois extension

Galois extension

In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...

 can be embedded into a Galois extension in such a way that the restriction map between the corresponding Galois group

Galois group

In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...

s is given.

Definition

Given a field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

 K and a finite group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 H, one may pose the following question (the so called inverse Galois problem

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 Q. This problem, first posed in the 19th century, is unsolved....

). 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

1. F is projective if and only if any finite embedding problem for F is solvable.
2. F is free of countable rank if and only if any finite embedding problem for F is properly solvable.