Every finite group admits a sufficiently large finite prime field

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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 k is termed sufficiently large with respect to a finite group G if the following are true:

  • The characteristic of k does not divide the order of G.
  • k contains d distinct d^{th} roots of unity, where d is the exponent of G. In other words, the polynomial x^d - 1 splits completely into linear factors over k.

Since the multiplicative group of a prime field is cyclic, a prime field with p elements is sufficiently large with respect to the finite group G iff the exponent of G divides p - 1. Similarly, since the multiplicative group of a finite field is cyclic, a finite field of order q = p^r is sufficiently large with respect to the finite group G iff the exponent of G divides q - 1.

Related facts

Facts used

  1. 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 m, there exist infinitely many primes p such that m|p-1.

Proof

By the definition of sufficiently large, it suffices to find a prime p such that p is congruent to 1 modulo the exponent of the group. The existence of such a prime is guaranteed by fact (1).