# 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

## Contents

## Definition

### Definition with symbols

A group is said to be **Galois-realizable** over a field if there exists a normal field extension of such that is the Galois group of .

If no field is specified, we assume that the field is , 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 .

### Definition in terms of the universal Galois group

A group is Galois-realizable over a field if it is the quotient of the universal Galois group of 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 :