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**Definition**

A ( extit{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 eq 0).
- D. The distributive law holds (a cdot (b + c) = ab + ac).

Note

Roughly, all of the above mean that you have notions of (+, -, imes, 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 ( extit{bits}). The rules for addition and multiplication are the usual ones save that $$1 + 1 = 0.]