Finite Fields: Definition, Properties, and Applications

In mathematics, a finite field is a field whose underlying set is finite. This article explores the definition, properties, and applications of finite fields, also known as Galois fields, in various areas of mathematics.

Definition of a Finite Field

A field is defined as a ring that contains a multiplicative identity (1), and the set of its nonzero elements forms an abelian group under multiplication. A finite field extends this concept to a finite set. Importantly, a finite ring with 1 whose nonzero elements form a group under multiplication is automatically a field, a fact proven by Wedderburn.

The classification of finite fields is straightforward. The cardinality (or order) of every finite field is a power of a prime number, denoted as (p^n), where (p) is a prime and (n) is a positive integer. This means every power of a prime occurs as the cardinality of a finite field. Moreover, given two finite fields of the same cardinality (p^n), they are isomorphic, indicating a deep structural similarity between them.

Properties of Finite Fields

Considering a finite field (GF(p^m)), if another field (GF(p^n)) with (0 leq m leq n) is embedded within it, the larger field is always a Galois extension of the smaller one. The Galois group of this extension is cyclic, meaning it can be generated by a single element.

The extension is also referred to as cyclotomic, obtained by adjoining roots of unity. This property underscores the deep connections between finite fields and complex algebraic structures, specifically through the group of invertible (n times n) matrices over finite fields, known as the general linear group (GL(n, GF(p))).

Construction of Finite Fields

The construction of finite fields is systematic and relies on the concept of galois fields or GF(p^n). These fields are denoted as (GF(p^n)) and can be explicitly represented:

GF(p) is built using elements that are all possible values of the form (1 1 ldots 1)((p) times). GF(p^n) is constructed using elements of the form (a0 a1x ldots an-1x^n-1), where the coefficients (a_i) are from (GF(p)), and the element (x) is a root of a specific polynomial, often a primitive polynomial.

Notably, all choices of a primitive polynomial yield isomorphic fields, meaning effectively, there is only one field (GF(p^n)).

Applications in Mathematics

The applications of finite fields are extensive and include representation theory, cryptology, coding theory, and algebraic geometry. In representation theory, finite fields are used to study the structure of groups via homomorphisms to the groups of invertible (n times n) matrices over finite fields. This approach has been instrumental in proving important theorems about groups.

Further applications include cryptographic systems, error-correcting codes, and algebraic geometry. For example, coding theory leverages the algebraic structure of finite fields to design robust error-correction codes, while algebraic geometry uses these fields in the study of algebraic varieties over finite fields.

In summary, finite fields, or Galois fields, are foundational in many areas of mathematics, from group theory to cryptology, and their study continues to enrich our understanding of algebraic structures and their applications.