Name given to an algebraic structure (K, ⊕ , ⊗) formed by a set K in which two operations denoted by ⊕ and ⊗ are second internal binary operation that satisfy the following conditions:
- (K, ⊕) forms a commutative group;
- (K*, ⊗) forms a group in which K* is composed of all the elements of K except for the neutral element of (K, ⊕);
- The operation ⊗ distributes over the operation ⊕.
Properties
- Commutative field
Field in which the second internal binary operation ⊗ is also commutative. - Ordered field
Field in which a total order can be defined that is compatible with each of the second internal binary operation ⊕ and ⊗.
Example
The sets \(\mathbb{Q}\) and \(\mathbb{R}\) which include the operations + and × are fields whose neutral elements are 0 and 1, respectively.