# 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 $G$.