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.
- The rational numbers , with operations the usual addition and multiplication.
- The real numbers , with operations the usual addition and multiplication.
- The complex numbers , with operations the usual addition and multiplication.
- The integers modulo p (denoted ), where p is prime. Here the operations are addition and multiplication modulo p. Observe that if p 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.
- The field of real numbers of the form , where both a and b are rational.
- Finite fields: for each prime number p and positive integer n, there is a unique (up to isomorphism) finite field of cardinality is pn. This field is of characteristic p.
- 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 .
- The p-adic fields and Ωp, which play a prominent role in number theory.
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.
Technically, a field F consists of a set of elements and two binary operations, addition + and multiplication *, which obey the following axioms:
1. Closure: If a,b ∈ F then a + b ∈ F.
2. Associativity: For a,b,c ∈ F, (a + b) + c = a + (b + c).
3. Commutativity: For a,b ∈ F, a + b = b + a
4. Identity: There exists an element 0 ∈ F such that a + 0 = 0 + a = a for all a ∈ F.
5. Inverse: If a ∈ F then there exists an (-a) ∈ F such that a + (-a) = (-a) + a = 0.
1. Closure: If a,b ∈ F then a*b ∈ F.
2. Associativity: For a,b,c ∈ F. (a*b)*c = a*(b*c).
3. Commutativity: For a,b ∈ F, a*b = b*a
4. Identity: There exists an element 1 ∈ F such that a*1 = 1*a = a for all a ∈ F.
5. Inverse: If a ∈ F and a ≠ 0 then there exists an a-1 ∈ F such that a*a-1 = a-1*a = 1.
For a,b,c ∈ F, a*(b + c) = a*b + a*c.
Sometimes the condition 0 ≠ 1 is also included.