# 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).