  ## Finite Fields

A set $F$, which is closed under two binary operations, which we denote by "$+$" and "$\cdot$", is called a field if it satisfies the following properties...

1. $F$ is associative with respect to addition:

For all $a,b,c \in F$, we have $a+(b+c) = (a+b)+c$

2. $F$ is commutative with respect to addition:

For all $a,b \in F$, we have $a+b=b+a$

3. There is an element in $F$ which we call the additive identity and denote by $0$ such that for every $a \in F$, we have $a + 0 = a$

4. For every $a \in F$, there exists an element $-a \in F$ which we call the additive inverse of $a$ such that $a + (-a) = 0$

5. $F$ is associative with respect to multiplication:

For all $a,b,c \in F$, we have $a \cdot (b \cdot c) = (a \cdot b) \cdot c$

6. $F$ is commutative with respect to multiplication:

For all $a,b \in F$, we have $a \cdot b = b \cdot a$

7. There is an element in $F$ which we call the multiplicative identity and denote by $1$ such that for every $a \in F$, we have $a \cdot 1 = a$

8. For every $a \in F$, there exists an element $a^{-1} \in F$ which we call the multiplicative inverse of $a$ such that $a \cdot a^{-1} = 1$

9. In $F$, multiplication distributes over addition in the usual way:
For all $a,b,c \in F$, we have $a \cdot (b + c) = a \cdot b + a \cdot c$

If a field $F$ contains only a finite number of elements, we say that $F$ is a finite field.