Field (mathematics)

A field is a commutative ring which contains a non-zero multiplicative identity and all non-zero elements have multiplicative inverses. Loosely, a field is a collection of entities with well-behaved and compatible addition and multiplication operations. A few examples serve to illustrate this point.

## Examples[edit]

1. The rational numbers , with operations the usual addition and multiplication.
2. The real numbers , with operations the usual addition and multiplication.
3. The complex numbers , with operations the usual addition and multiplication.
4. The integers modulo  (denoted ), where  is prime. Here the operations are addition and multiplication modulo . Observe that if  is not prime, then  is not a field. For example, the element  has no multiplicative inverse modulo 6! In this case,  has only the structure of a ring.
5. The field  of real numbers of the form , where both  and  are rational.
6. Finite fields: for each prime number  and positive integer , there is a unique (up to isomorphism) finite field of cardinality is . This field is of characteristic .
7. The set of meromorphic functions on a complex manifold, with pointwise addition and multiplication. For example, the set of meromorphic functions on  or the unit disk .
8. The p-adic fields  and , which play a prominent role in number theory.

The characteristic of a field must be either 0 or a prime number p. A field of characteristic 0 is necessarily infinite.

Fields play an important role in nearly every area of mathematics, and are one of the most basic objects studied by algebra. The study of the relationships between different fields, and in particular subfields of a given field, leads to the study of Galois theory, and makes possible the proof of Abel's theorem and was one of the motivations for the early study of fields and abstract algebra more generally.

## Axioms[edit]

Technically, a field  is a set of elements endowed with two binary operations,  and  (with properties analogous to addition and multiplication, respectively), which obey the following axioms:

* Addition axioms
1. Closure: If  then .
2. Associativity: For , .
3. Commutativity: For  .
4. Identity: There exists an element  such that  for all .
5. Inverse: For all  there exists an element  such that .
* Multiplication axioms
1. Closure: If  then .
2. Associativity: For , .
3. Commutativity: For , .
4. Identity: There exists an element  such that  for all .
5. Inverse: If  and  then there exists an element  such that .
* Distributivity: For , .

Sometimes the condition 0 ≠ 1 is also included.