
# B: Fields

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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 $$1 + 1 = 0.$$