(Untitled)

[littlejohnny]

Now that I think of it, the distributive property may be a requirement for a set of elements to form a field or a ring, or maybe both. But how do you prove that the real numbers form a field, or the integers a ring?

Anyone? Anyone?

......Clint?

Comments

[balachthon]

The distributive property is a requirement for both rings and fields. But a ring R can be noncommutative, so in fact the distributive property for rings states that for all a,b,c in R:
a(b+c) = ab + ac AND (b+c)a = ba + ca.

How you prove that the integers or reals are a ring or field (respectively) depends on how you define/construct the integers or reals, and what you take to be given or not. For most purposes, it's fine just to say, for instance, "The reals are by definition a complete ordered field," and then work out their properties from there. Of course, technically you'd have to prove--using abstract algebra which relies ultimately on set theory--that a complete ordered field exists, and that any two complete ordered fields are isomorphic to one another--but other people have already done that, so you don't really need to worry about it, right? Right.

[balachthon]

This is by far the nerdiest post I have ever made.