linguistic:paraphrasing, structural:expansion
Finite fields

Finite fields (also called Galois fields) are fields with finitely many elements, whose number is also referred to as the order of the field. The above introductory example F is a field with four elements. Its subfield F is the smallest field, because by definition a field has at least two distinct elements, 0 and 1.

The simplest finite fields, with prime order, are most directly accessible using modular arithmetic. For a fixed positive integer n, arithmetic "modulo n " means to work with the numbers

The addition and multiplication on this set are done by performing the operation in question in the set Z of integers, dividing by n and taking the remainder as result. This construction yields a field precisely if n is a prime number. For example, taking the prime n = 2 results in the above-mentioned field F. For n = 4 and more generally, for any composite number (i.e., any number n which can be expressed as a product n = r ⋅ s of two strictly smaller natural numbers), Z / n Z is not a field: the product of two non-zero elements is zero since r ⋅ s = 0 in Z / n Z, which, as was explained above, prevents Z / n Z from being a field. The field Z / p Z with p elements (p being prime) constructed in this way is usually denoted by F.

Every finite field F has q = p elements, where p is prime and n ≥ 1. This statement holds since F may be viewed as a vector space over its prime field. The dimension of this vector space is necessarily finite, say n, which implies the asserted statement.

A field with q = p elements can be constructed as the splitting field of the polynomial

Such a splitting field is an extension of F in which the polynomial f has q zeros. This means f has as many zeros as possible since the degree of f is q. For q = 2 = 4, it can be checked case by case using the above multiplication table that all four elements of F satisfy the equation x = x, so they are zeros of f. By contrast, in F, f has only two zeros (namely 0 and 1), so f does not split into linear factors in this smaller field. Elaborating further on basic field-theoretic notions, it can be shown that two finite fields with the same order are isomorphic. It is thus customary to speak of the finite field with q elements, denoted by F or GF(q).