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 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

Relation with other properties

Weaker properties