Sufficiently large field

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

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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 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.

Relation with other properties

Weaker properties

• Character-determining field
• Character-separating field
• Class-determining field
• Class-separating field
• Splitting field: For full proof, refer: Sufficiently large implies splitting