Field (mathematics)
Definition and Basics
In Mathematics, a field is a set equipped with two operations that generalize the arithmetic operations of addition and multiplication. As such, a field can be thought of as consisting of numbers. The operations are required to satisfy several properties that are familiar from the arithmetic of rational numbers and real numbers. The most important of these properties are commutativity, associativity, distributivity, and the existence of additive and multiplicative identities and inverses.
Structure
A field is a set F together with two operations called addition and multiplication. An operation is a function that takes two elements of the set and produces another element of the set. The result of the addition of a and b is called the sum of a and b, and is denoted a + b. Similarly, the result of the multiplication of a and b is called the product of a and b, and is denoted ab or a·b.
Properties
Fields have several fundamental properties that make them a key structure in many areas of mathematics. These properties are often used to define fields in the first place, and they are what make fields useful in practice.
Closure
The set F is closed under the operations of addition and multiplication. This means that the sum or product of any two elements in F is also an element in F.
Associativity
The operations of addition and multiplication are associative. This means that for any elements a, b, and c in F, the equation (a + b) + c = a + (b + c) holds for addition, and the equation (ab)c = a(bc) holds for multiplication.
Commutativity
The operations of addition and multiplication are commutative. This means that for any elements a and b in F, the equation a + b = b + a holds for addition, and the equation ab = ba holds for multiplication.
Identity elements
There are two special elements in F called the additive identity and the multiplicative identity. The additive identity is an element 0 such that for any element a in F, the equation a + 0 = a holds. The multiplicative identity is an element 1 (distinct from 0) such that for any element a in F, the equation a·1 = a holds.
Inverse elements
Every element in F has an additive inverse and every non-zero element in F has a multiplicative inverse. The additive inverse of an element a is an element -a such that a + (-a) = 0. The multiplicative inverse of a non-zero element a is an element a^-1 such that a·a^-1 = 1.
Distributivity
The operations of addition and multiplication are related by the distributive law. This means that for any elements a, b, and c in F, the equation a·(b + c) = a·b + a·c holds.
Examples
Fields are ubiquitous in mathematics. Here are some examples:
Rational numbers
The set of all rational numbers, denoted by Q, forms a field with the usual operations of addition and multiplication.
Real numbers
The set of all real numbers, denoted by R, forms a field with the usual operations of addition and multiplication.
Complex numbers
The set of all complex numbers, denoted by C, forms a field with the usual operations of addition and multiplication.
Finite fields
There are also fields that have only a finite number of elements. These are called finite fields or Galois fields.
Applications
Fields are used in many areas of mathematics and its applications. Here are some examples:
Algebra
In Algebra, fields are used to construct and study algebraic structures such as rings, vector spaces, and algebraic varieties.
Number theory
In Number Theory, fields are used to study properties of numbers, particularly in the context of algebraic number theory and arithmetic geometry.
Geometry
In Geometry, fields are used to define and study geometric objects, particularly in the context of algebraic geometry and differential geometry.
Cryptography
In Cryptography, finite fields play a crucial role in the design of many cryptographic algorithms and protocols.