the definition of field in mathematics

Question

Every textbooks can tell, that a field is a set with two operations called addition and multiplication, which satisfy particular axioms (ordinarily divided into three parts: axioms for addition, for multiplication, and the distributive law).
According to what we have learned in elementary math, obviously the real field $\mathbb{R}$ satisfies every axioms. But, as the defination of a concept that has general value, the axioms seems too complicated and not that 'natural' as other defination of fundamental concept.
In detail, the axioms for each operation has five lines, called closure, communtative law, associative law, unit element and inverse element. But the confusion which comes from their source and complexity exists still.
So, my question are, where are the axioms 'come' from? Why five laws can define an operation and we need to add the distributive law to perfect the defination of field?
p.s. I guess my question may solve after reading some algebra because I have never read any professional algebra book yet. You can recommend me to read particular parts of algebra which help me to answer the questions by myself. THX!

Answer 1

The justification for the field axioms came from the fact that various important structures in mathematics - including the rational numbers, the real numbers, the complex numbers, and integers modulo a prime $p$ - all had certain features in common, and one way to codify those features was via the field axioms. This is a very common practice in mathematics, which is:

1. Notice that several interesting structures share certain properties.

2. Try to define a general structure that captures those properties.

3. See what can be proven about that general structure, which then doesn't need to be proven individually for each example any more.

For fields in particular, the axioms actually have a nice "reduction" if you already know about another kind of general structure, namely groups. If you know what a group is, then you can define a field like this:

A field is a set $F$ along with two operations $+$ and $\times$, both commutative, such that:

1. $(F, +)$ is an Abelian group.
2. If $0$ is the identity of $(F, +)$, then $(F\setminus\{0\}, \times)$ is an Abelian group.
3. $\times$ distributes over $+$.

Alternatively, there's a structure called a ring, and a field is just a ring with some extra restrictions - mostly that everything that isn't zero has a multiplicative inverse, which turns out to be a really useful property in a lot of instances.