Finite Fields, A Unique Field of Order p

A Unique Field of Order p

If a field has p elements (p prime), call the additive identity 0 and the multiplicative identity 1. Addition defines a group, and there is only one group of order p, hence addition in the field is given by addition mod p. Let 1 generate this group. Multiplication is completely determined by the distributive property, and agrees with multiplication mod p. Since every nonzero element has an inverse mod p, we have a division ring, and a field. for every prime p, there is a unique field of order p.

Note that the integers mod n, for n composite, do not form a field, and cannot be part of a field, since this would admit zero divisors a×b = n = 0.

What is the order of a larger field, a field containing Z_p?

Generate an additive cycle by repeatedly adding any nonzero element x to itself. This is the same as 1x, 2x, 3x, etc. In a field based on Z_p, all such additive cycles must have length p. If a finite field contains n elements, and q is a prime that divides n, the additive group of the field contains a subgroup of order q by Macay's theorem. Yet q must equal p, hence the order of any finite field is a power of its prime characteristic p.