# Field

## Field

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:

1. (K, ⊕) forms a commutative group;
2. (K*, ⊗) forms a group in which K* is composed of all the elements of K except for the neutral element of (K, ⊕);
3. 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.