# The field axioms

A system of numbers in mathematics is called a field and is characterised by the following axioms.

We assume that we have a set, the elements of which we call numbers. For any two numbers $x,y$ the expressions $x+y$ and $x\cdot y$ also denote numbers.

- There is a special number, written as 0, for which we have
$$0+x = x$$ for all numbers $x$. The number $0$ is called the neutral element of $+$.
- There is a special number, written as 1, for which we have
$$1\cdot x= x$$ for all numbers $x$. The number $1$ is called the neutral element of $\cdot$.
- $0$ is different from $1$.
- For all $x$ there is a number, written as $-x$, for which we have
$$ x+(-x) = 0$$
- For all $x\not=0$ there is a number, written as $x^{-1}$ or $\frac 1 x$ or $1/x$, such that $$x\cdot x^{-1} = 1$$
- We have the usual rules of commutativity and associativity
\begin{align}
x+y& =y+x\\
x\cdot y &= y\cdot x\\
(x+y)+z &=x+(y+z)\\
(x\cdot y)\cdot z & = x\cdot (y\cdot z)
\end{align}

- The law of distributivity $$x\cdot (y+z)= x\cdot y + x\cdot z$$

**Remark:** We see here that mathematicians like to treat minus and division slightly differently from how we learned it at school. As you can see, minus is not an operation we apply to two numbers, but only to one number. But we can still write $x-y$ with the same meaning as before, as long as we understand that this is an abbreviation for $x+(-y)$. Division is treated in the same way.