*$F$ is associative with respect to addition:*

For all $a,b,c \in F$, we have $a+(b+c) = (a+b)+c$*$F$ is commutative with respect to addition:*

For all $a,b \in F$, we have $a+b=b+a$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 $a + 0 = a$For every $a \in F$, there exists an element $-a \in F$ which we call

*the additive inverse of $a$*such that $a + (-a) = 0$*$F$ is associative with respect to multiplication:*

For all $a,b,c \in F$, we have $a \cdot (b \cdot c) = (a \cdot b) \cdot c$*$F$ is commutative with respect to multiplication:*

For all $a,b \in F$, we have $a \cdot b = b \cdot a$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 $a \cdot 1 = a$For every $a \in F$, there exists an element $a^{-1} \in F$ which we call

*the multiplicative inverse of $a$*such that $a \cdot a^{-1} = 1$*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**.