To prove that a field is complex

[friend]

I understand that the complex numbers form a "field" since the complex numbers are closed under addition, subtraction, multiplication, and division. And I understand the complex numbers are not an ordered field since it's not possible to define a relation z1<z2.

My question is: Are all not ordered fields necessarily complex? Then how would you prove that a field is not ordered, is that something that is observed in a system or imposed? Thanks.

 

[micromass]

Take F={0,1} with

$$0+0=1+1=0,~1+0=0+1=1$$

and

$$0*0=1*0=0*1=0,~1*1=1$$

then F is a field that can not be ordered.

 

[DonAntonio]

I understand that the complex numbers form a "field" since the complex numbers are closed under addition, subtraction, multiplication, and division. And I understand the complex numbers are not an ordered field since it's not possible to define a relation z1<z2.

My question is: Are all not ordered fields necessarily complex?

No. Any finite field is not orderable (in fact, any field of positive characteristic is not ordered), or any non-real extension of $\,\mathbb{Q}\,$ is not orderable...

Then how would you prove that a field is not ordered, is that something that is observed in a system or imposed? Thanks.

A field can be ordered iff -1 can't be expressed as a sum of squares, or equivalently iff a sum of squares equals zero iff every summand is zero.

DonAntonio

 

[friend]

Take F={0,1} with

$$0+0=1+1=0,~1+0=0+1=1$$

and

$$0*0=1*0=0*1=0,~1*1=1$$

then F is a field that can not be ordered.

This seems like a very strange way to define + and *. Are you saying that in a field that we can define + and * and way we wish? Or is there some requirements for + and * so that they are consistent with each other?

 

[Sankaku]

Are you saying that in a field that we can define + and * and way we wish?

A field is a specific algebraic structure with its own axioms so, no, we can't do anything we wish. What Micromass described is a special (very small) field.

http://en.wikipedia.org/wiki/Field_(mathematics )

I would suggest doing a little reading on Groups and Rings as well, to give Fields some context. Wikipedia might not be the best place for a beginner to start. Try a free textbook like this:

http://abstract.ups.edu/

 

[spamiam]

This seems like a very strange way to define + and *.

It's not that strange at all. It's just like a clock with only two hours: 0 and 1. Take a look at this article: http://en.wikipedia.org/wiki/Modular_arithmetic

Are you saying that in a field that we can define + and * and way we wish? Or is there some requirements for + and * so that they are consistent with each other?

No, there are axioms that + and * must satisfy in order for (F, +, *) to be considered a field. Briefly, (F, +) must be an abelian group, $(F^\times, *)$ must be an abelian group and the distributive law must hold. You can read the axioms in more detail here: http://en.wikipedia.org/wiki/Field_(mathematics).