Undergrad Are there relations that are valid for any field?

[SVN]

I use the word «field» in purely algebraic sense here. Sometimes, when reading textbooks I encounter sentences like «Although the formulae in this section derived for the field of real numbers, they remain valid for complex numbers field as well». Or even more general variant of it: «...remain valid for any field» (of course, I am not talking here about standard operations like addition, multiplication and so on, which make it possible to for a set to be called a field). Frankly, I can't give you an example of such textbooks right now, but I feel quite sure that I encountered such phrases not once (just did not try to ponder them over than). What does it mean for a relation «to be valid for any field»?

And the opposite question in some sense: obviously, different fields have much in common, but they also possesses some «fine internal structure» that makes them qualitatively different from each other. Is it possible to formalise and understand those structure differences? OK, the complex numbers field does behave differently from its real numbers counterpart, but it is just a statement, a fact, but why it is so, can the differences be predicted before studying behaviour of both fields? For example, can it be, that differences between real and complex analyses stem from the fact one of these fields is ordered while the other is not?

 

[fresh_42]

When it comes to fields, the major criteria are algebraic closure and characteristic.

As e.g. $\mathbb{R}$ isn't algebraic closed, it happens that some eigenvalues are not part of the field, whereas in $\mathbb{C}$ they are. The amount of values available is essential to many theories. Also real and complex differentiation are quite different, and not only because of the different real dimension.

The characteristic is also important. E.g. many things do not apply to characteristic $2$ fields: no minus!
An easy example is the commutator relation. If $\operatorname{char} \mathbb{F} \neq 2$ then
$$[X,X] = 0 \Longleftrightarrow [X,Y] = -[Y,X]$$
This is not true in the other case. So often especially the case of characteristic $2$ is ruled out. But of course there are other differences, too, for finite characteristics, e.g. the Frobenius homomorphism.

Statements at which the remark "valid for any field" is attached, usually only use the fact that scalars can be divided.

 

[mathman]

"field" in algebra has a very precise definition, where the four principal math. operations are defined.
https://en.wikipedia.org/wiki/Field_(mathematics)

 

[SVN]

@fresh_42 Thank you for your tip, I'll have to ponder over it.

@mathman It is absolutely true, but I fail seeing how it answers my question(s).

 

[Stephen Tashi]

obviously, different fields have much in common, but they also possesses some «fine internal structure» that makes them qualitatively different from each other.

There are fields that have only a finite number of elements, so I wouldn't say the internal structure of a field is necessarily "fine".

Is it possible to formalise and understand those structure differences?

The formal way to distinguish among things that satisfy a common set of axioms is to come up with additional axioms that rule out some of the things - and to continue adding axioms and ruling out more and more things until we are happy with what's left. If you want to do this with the collection of fields, one attempt is to add axioms that specify the topology of the field and that also say that the operations of addition and multiplication are continuous with respect to that topology. However, I myself have never read a book that took such an approach. (It would the sort of book that a Bourbaki would write.) It raises some deep questions. For example, in addition to topology, do we have to bring in the concepts of bases and dimension in order to distinguish among the fields we want to talk about?

If an article shows something is true for the real numbers and then claims it is true "for any field", it can be a challenging exercise to actually verify that fact. For example, if the material deals with calculus, we have to think about what limits, derivatives and integrals are in the case of finite fields.

 

[fresh_42]

(It would the sort of book that a Bourbaki would write.)