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.