# Sufficiently large field

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

- contains all the roots of unity, where is the exponent of .
- The polynomial splits completely over where is the exponent of .
- is a splitting field for every subgroup of .
- is a splitting field for every subquotient of .

### 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.