Beginner Analysis Question: Are The Field Axioms Any More Than Symbolic Relations

Question

I'm reading baby rudin for some mathematical exposure and have enjoyed surviving until page 5.

When I think about the definition of fields I see that they must obey certain axioms for addition and multiplication.

I am having trouble wrapping my head around whether these axioms actually have any intuitive meaning or are really just symbolic manipulation rules.

When you want to reason about the results of these axioms, are you reasoning about the operations they imply or on their very rigidly defined symbolic manipulation rules.

I think of multiplication which until now has held some intuitive meaning. Now re: this new definition of multiplication: does it need to mean anything to be used – is the notion of multiplication anything more than the symbolic manipulation rules. Anytime I've ever done multiplication in the past, have I just been using a precomputed cognitive heuristic to the formal symbolic derivation of the result through continuous application of the multiplication axioms?

Answer 1

"Is the notion of multiplication anything more than the symbolic manipulation rules?" Yes, it is. Most examples of fields have a multiplication which is the one we know. On the other hand, it is sometimes more general. Consider for example a finite field $\mathbb{F}_q$ for a prime power $q=p^n$ with $n\ge 2$. Then we can no longer view it like $\mathbb{Z}/q$, i.e., the multiplication is not given by multiplication of integers.

Answer 2

A non-rigorous definition of a field is "something that has addition and multiplication like the rationals, real numbers, or complex numbers". What do I mean by "addition and multiplication like..."? What these things have in common is the order doesn't matter, there's a distributive property, and there are multiplicative and additive inverses. Let's call a field anything that satisfies these properties. It turns out there are LOTS of other fields out there.

Anything you can prove about the reals, rationals, and complex numbers that use specifically the addition/multiplication rules will also be true for any other field. It doesn't matter if you're doing rational multiplication and rational addition or complex multiplication and complex addition, there's something common going on. That's the point of an abstract definition of fields.