# B: Fields

**Definition**

A \(\textit{field}\) \(\mathbb{F}\) is a set with two operations \(+\) and \(\cdot\), such that for all \(a, b, c \epsilon \mathbb{F}\) the following axioms are satisfied:

- A1. Addition is associative \((a + b) + c = a + (b + c)\).
- A2. There exists an additive identity \(0\).
- A3. Addition is commutative \(a + b = b + a\).
- A4. There exists an additive inverse \(-a\).
- M1. Multiplication is associative \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).
- M2. There exists a multiplicative identity \(1\).
- M3. Multiplication is commutative \(a \cdot b = b \cdot a\).
- M4. There exists a multiplicative inverse \(a^{-1}\) if \(a \neq 0\).
- D. The distributive law holds \(a \cdot (b + c) = ab + ac\).

Note

Roughly, all of the above mean that you have notions of \(+, -, \times, \div\) just as for regular real numbers.

Fields are a very beautiful structure; some examples are rational numbers \(\mathbb{Q}\), real numbers \(\mathbb{R}\), and complex numbers \(\mathbb{C}\). These examples are infinite, however this does not necessarily have to be the case. The smallest example of a field has just two elements, \(\mathbb{Z}_2 = {0, 1}\) or \(\textit{bits}\). The rules for addition and multiplication are the usual ones save that $$1 + 1 = 0.$$

### Contributor

David Cherney, Tom Denton, and Andrew Waldron (UC Davis)