Sufficiently large field

Definition
Let $$G$$ be a finite group and $$k$$ a field. We say that $$k$$ is sufficiently large for $$G$$ if the characteristic of $$k$$ does not divide the order of $$G$$, and the following equivalent conditions are satisfied:


 * 1) $$k$$ contains all the $$m^{th}$$ roots of unity, where $$m$$ is the exponent of $$G$$.
 * 2) The polynomial $$x^m - 1$$ splits completely over $$k$$ where $$m$$ is the exponent of $$G$$.
 * 3) $$k$$ is a defining ingredient::splitting field for every subgroup of $$G$$.
 * 4) $$k$$ is a splitting field for every subquotient of $$G$$.

Equivalence of definitions

 * The equivalence of definitions (1) and (2) is straghtforward field theory.
 * For (1) implies (4), refer sufficiently large implies splitting for every subquotient.
 * (4) implies (3) is clear.
 * For (3) implies (1), refer splitting field for every subgroup implies sufficiently large.

Weaker properties

 * Character-determining field
 * Character-separating field
 * Class-determining field
 * Class-separating field
 * Splitting field: