Jtdylktuy

High school student here...

I am currently working on Spivak and I'm not very far (just reached questions in chapter 1). I was discussing my progress with someone recently and they asked me to explain why $Bbb C$ can't be an ordered field like $Bbb R$. I had recently read Spivak's proof that the square of any nonzero number is always positive, so after some thinking, I concluded that $Bbb C$ can't be an ordered field because $i^2 = -1$. The person who asked me the question told me I was correct and at first I was very happy. Then I realized I still don't know what the heck an ordered field is. I looked it up on Wikipedia and I couldn't really get an idea of what the article meant. Spivak doesn't cover fields until chapter 27 and I feel like I wouldn't really get it by skipping there. Could someone give a layman-esque overview of what an ordered field is and why $Bbb R$ is one really?

You should have a couple of examples in your mind throughout this post (and generically). A field is roughly something that behaves like the rationals $mathbb{Q}$, the reals $mathbb{R}$, or the complex numbers $mathbb{C}$. There are other fields, but these are the ones which you are probably most familiar with.

More generally, a field is a set of things in which it makes sense to add two things together and multiply two things together. Both of these operations should be commutative, meaning that $a+b = b+a$ and $ab = ba$. Further, there is a distributive law connecting addition and multiplication, so that you have $a(b+c) = ab + ac$.

A field should also have two distinguished elements, that we might suggestively call $0$ and $1$, so that $a+0 = a$ for all $a$ and $ b*1 = 1$ for all $b neq 0$.

Finally, for every $b neq 0$, there should be another element $b^{-1}$ such that $bb^{-1} = 1$.

In other words, a field is a set where addition and multiplication work essentially as they work in $mathbb{R}, mathbb{Q}$, or $mathbb{C}$.

Now one should note that not all fields really do look like $mathbb{R}, mathbb{Q}$, or $mathbb{C}$. Some fields have could have weird definitions of "multiplication" or "addition" --- what's important isn't that they're exactly the same, but that they behave in ways similar to the fields we're very familiar with.

Returning to the question at hand, we now know what a field is. An ordered field is a field where you can say, for any two elements $a$ and $b$, whether $a < b$, $a = b$, or $a > b$. And this should be in such a way that the typical ideas we have about inequalities hold true. These ideas include

• $1 > 0$


• if $a > b$ and $b > c$, then $a > c$
  


• if $a > 0$ and $b > c$, then $ab > ac$
  

and things of that nature.