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!
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11$\begingroup$ You are "putting the cart before the horse". Specific important fields came first, and the field axioms came later. Mainly to prove general theorems. $\endgroup$– SomosAug 15 at 18:30
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3$\begingroup$ The rationals are, perhaps, a more natural example. Certainly they are more natural from an algebraic perspective. All the axioms do is to formalize the basic arithmetic in the rationals. We have commutative addition and multiplication with inverse operations, and these distribute over each other. That's it! Of course, then it turns out that there are many other interesting examples, even some finite examples. $\endgroup$– luluAug 15 at 18:31
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2$\begingroup$ It's worth noting that fields are just one example of many algebraic structures that mathematicians are interested in studying. You don't have to think of them as being a fundamental concept – rather, these structures are ubiquitous enough to deserve a name. And there are important structures that have fewer defining axioms, such as groups. Generally speaking, groups bear far less resemblance to "number systems" than fields do. $\endgroup$– JoeAug 15 at 18:49
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2$\begingroup$ Mathematicians have produced systems with far more complicated definitions than the definition of field. For instance, Lie algebra, locally compact topological space, triangulated category, etc. Particular definitions are justified by the existence of interesting examples and general theorems that use them. There are lots of examples of fields of different sorts (finite fields, ray class fields, fields of meromorphic functions etc) and lots of amazing theorems about them. $\endgroup$– Barry SmithAug 15 at 20:13
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1$\begingroup$ As an aside, all definitions are complicated! You state that a field is a set with addition and multiplication satisfying some "complicated" axioms. In response I'd like to ask you: what is the definition of a set? Pretty much every definition in mathematics, unspooled enough, is horrendous--- we simply abstract away the horror and are left with the intuition which (hopefully) is captured by the definition. $\endgroup$– WilliamAug 16 at 1:04
1 Answer
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:
Notice that several interesting structures share certain properties.
Try to define a general structure that captures those properties.
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:
- $(F, +)$ is an Abelian group.
- If $0$ is the identity of $(F, +)$, then $(F\setminus\{0\}, \times)$ is an Abelian group.
- $\times$ 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.
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$\begingroup$ This is a very good answer. I think that the part of the problem is that many textbooks present algebraic structures in a way that is historically backwards. First, they give the definition of the structure, and then they give examples, sometimes as if this is an afterthought. The approach found in textbooks does have some merits: for instance, as your answer mentions, it means that you can quickly establish theorems that apply to all of the examples at once, rather than proving them individually. $\endgroup$– JoeAug 16 at 10:38
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$\begingroup$ However, if one fails to mention how fields actually came to be, then the motivation for why we are interested in them goes missing. Students are left wondering why we are interested in these seemingly arbitrary structures. But we are interested in fields precisely because they appear naturally in a range of different areas of mathematics, and it was only after we noted their appearance that we formulated the field axioms. $\endgroup$– JoeAug 16 at 10:38