Every finite group admits a sufficiently large finite prime field
Contents
Definition
For any finite group, there exists a prime field (not of characteristic zero) that is sufficiently large with respect to the finite group.
Definitions used
Sufficiently large field
Further information: sufficiently large field
A field is termed sufficiently large with respect to a finite group
if the following are true:
- The characteristic of
does not divide the order of
.
-
contains
distinct
roots of unity, where
is the exponent of
. In other words, the polynomial
splits completely into linear factors over
.
Since the multiplicative group of a prime field is cyclic, a prime field with elements is sufficiently large with respect to the finite group
iff the exponent of
divides
. Similarly, since the multiplicative group of a finite field is cyclic, a finite field of order
is sufficiently large with respect to the finite group
iff the exponent of
divides
.
Related facts
Facts used
- There are infinitely many primes that are one modulo any modulus: This is the easy case of Dirichlet's theorem on primes in arithmetic progressions, which states that given any positive integer
, there exist infinitely many primes
such that
.
Proof
By the definition of sufficiently large, it suffices to find a prime such that
is congruent to
modulo the exponent of the group. The existence of such a prime is guaranteed by fact (1).