Galois-realizable group

This term associates to every field, a corresponding group property. In other words, given a field, every group either has the property with respect to that field or does not have the property with respect to that field

This term is related to: Galois theory
View other terms related to Galois theory | View facts related to Galois theory

Definition

Definition with symbols

A group $G$ is said to be Galois-realizable over a field $K$ if there exists a normal field extension $L$ of $K$ such that $G$ is the Galois group of $L/K$.

If no field is specified, we assume that the field is $K = \mathbb{Q}$, the field of rational numbers.

The inverse Galois problem essentially asks for which finite groups can be expressed as the Galois group of a field extension of $\mathbb{Q}$.

Definition in terms of the universal Galois group

A group $G$ is Galois-realizable over a field $K$ if it is the quotient of the universal Galois group of $K$ by a normal subgroup which is closed under the naturally given Krull topology.

Relation with other properties

Weaker properties

The following properties are known to be weaker than the property of being Galois-realizable over $\mathbb{Q}$: