Constructing Finite Fields by Adding an Imaginary Unit: Mathematical Foundations and Ethereum Implementation

Let p be a prime number. Then the integers mod p form a finite field.

The number of elements in a finite field must be a power of a prime, i.e. the order q = p^n for some n.

When n > 1, we can take the elements of our field to be polynomials of degree n − 1 with coefficients in the integers mod p.

Addition works just as you'd expect addition to work, adding coefficients mod p, but multiplication is a little more complicated.

You multiply field elements by multiplying their polynomial representatives, but then you divide by an irreducible polynomial and take the remainder.