Axioms of a Field

The following axioms must be proven for a Field to be defined. Most definitions of a field use addition ($+$) and multiplication ($\times$) as their binary operators, because it’s easy to understand how they work.

Assume $F$ is the set that which the field to be defined uses.

F0. Closure under Addition & Multiplication §

$$\forall x,y\in F,x\times y\in F$$

The two axioms above can be inferred from the definition of a binary operator.

F1. Commutativity of Addition §

F2. Associativity of Addition §

$$\forall x,y,z\in F,x+(y+z)=(x+y)+z$$

F3. Additive Identity §

\exists \ 0_{F} \in F: \forall \ x \in F: x + 0_{F} = 0_{F} + x = x$$ $0_{F}$ is denoted to be the **field zero**. Note that $x$ can also be $0_{F}$ [!warning] Different zeroes! The $0_{F}$ element here is *not* the number zero. Because mathematicians like to be confusing, the additive identity is 0, which is the same symbol as the number zero. But in another field, for example, 0 can still be the identity, but the operator could be something completely different. That's why I prefer to use $\text{i}$ or $\scr i$ as the field identity. ##### F4. Additive Inverses

\forall \ x \in F: \exists \ x’ \in F: x + x’ = 0_{F} = x’ + x

A lot of textbooks and internet sites denote the additive inverse as $-x$. I feel like it causes you to associate it with normal subtraction, which isn't always the case. ##### F5. Commutativity of Multiplication

\forall \ x,y \in F, \ \ \ x \times y = y \times x

##### F6. Associativity of Multiplication

\forall \ x,y,z \in F, \ \ \ x \times (y \times z) = (x\times y) \times z

##### F7. Multiplicative Identity

\exists \ 1_{F} \in F: \forall \ x \in F: x \times 1_{F} = 1_{F} \times x = x

$1_{F}$ is also called the **field unity** of $F$ ##### F8. Multiplicative Inverses

\forall \ x \in F: \exists \ x^{-1} \in F: x \times x^{-1} = 1_{F} = x^{-1} \times x

##### F9. Distributivity of Multiplication and Addition

\forall \ x,y,z \in F: x\times(y+z) = (x\times y) + (x\times z)

##### F10. Distinct Additive and Multiplicative Identities

0_{F} \neq 1_{F}

Now, because we have the additive inverse ($x'$) and multiplicative inverse ($x^{-1}$), we can define *new* binary operators: subtraction and division:

a -b = a + (b’)

a \div b = a \times b^{-1}$$
And that basically stems down to how a field is defined. If you have a set of anything, and you can show that the 10 field axioms are proved, that set can be called a field.