Splitting field

In terms of realization of irreducible representations
A splitting field for a finite group is a field satisfying both the following two conditions:


 * 1) Every linear representation of the group is completely reducible. This is equivalent to saying that the characteristic of the field does not divide the order of the group.
 * 2) Any linear representation of the group over any extension of the given field, is equivalent to a linear representation over the field itself

In terms of the character ring
A splitting field for a finite group is a field satisfying both the following two conditions:


 * 1) Every linear representation of the group is completely reducible. This is equivalent to saying that the characteristic of the field does not divide the order of the group.
 * 2) The character ring of the group over the field is equal to the character ring of the group over any extension of the field. Here, the character ring of a group over a field is the ring of $$\Z$$-linear combinations of characters of representations of the group realizable over that field.

Note that in some alternative definitions, only condition (2) is imposed for being a splitting field, thus also including the modular case where not all representations are completely reducible.

Stronger properties

 * Weaker than::Sufficiently large field:
 * Weaker than::Minimal splitting field: A splitting field not containing any smaller splitting field.

Weaker properties

 * Stronger than::Character-separating field
 * Stronger than::Class-separating field