# What is a finite field

1. Jul 23, 2014

### Greg Bernhardt

Definition/Summary

All finite fields are known; they are the Galois fields GF(p^n), where p is a prime.

They have addition group Z(p)^n and multiplication group Z(p^n-1); their multiplication groups are cyclic.

If p = 2, then addition and multiplication can be done very fast by typical computer hardware using bitwise exclusive or and shifting.

Equations

Extended explanation

The finite fields GF(p) are {0, 1, ..., p-1} under addition and multiplication modulo p, which must be a prime number.

Finite fields GF(pn) for n > 1 can be described using polynomials in a variable x with coefficients having values in {0, 1, ..., p-1}.

Every element is a polynomial with a degree at most n-1.

Element addition is polynomial addition modulo p, while element multiplication is polynomial multiplication modulo p and a degree-n primitive or irreducible polynomial.

A primitive polynomial is one that cannot be factored in this construction of GF(pn). Primitive polynomials are not unique; there are
$N(p,n) = \frac{1}{n} \sum_{m|n} \mu(m) p^{n/m}$

monic ones, where μ is the Möbius mu function. That function is (-1)number of prime factors if they all have power 1, and 0 otherwise.

A finite field GF(pn) has subfield GF(pm) if m evenly divides n. If n is a prime, then GF(pn) only has only the trivial subfields, itself and GF(p).

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