Demystifier said:
To be consistent with what I said before, I would say that analytic number theory and algebraic number theory are not two branches but two approaches within the same branch. (Alternatively, one could also say that they are two sub-branches, and stipulate that a sub-branch is not a branch.)
But one should not forget that the goal of categorization of math is not to establish a truth. The goal is merely practical, e.g. when you want to organize a lot of math books and papers into directories and sub-directories at your computer. There is no perfect categorization, but some categorization scheme simply needs to be chosen for practical reasons. The scheme I have described works very well for me. Someone else can choose a different scheme, and if it works well for him/her, that's fine too.
There are many practical reasons for classifying things and they can be completely personal.
But let's take your example of "basic number theory", propositions about numbers that can be stated without reference to analysis or other fields.
For instance the Fundamental Theorem of Algebra.
There have been many proofs of the FTA all of which have used something other than basic number theory. The ones I have seen use Complex Analysis, General Topology, or Differential Topology. The Complex Analysis proofs and the Differential Topology proof demonstrate that every complex polynomial determines a surjective map from the Riemann sphere onto itself. One can therefore view the FTA as a statement about mappings of the Riemann sphere. The more algebraic proofs still use topology since they rely on the intermediate Value Theorem. There is also one of Gauss's proofs that the two surfaces determined by the real and complex parts of a polynomial must have non-empty intersection. I am not sure how this proof works.
One might ask whether there are"purely algebraic" proofs and it seems that maybe there are. The following synopsis describes an analysis free proof but requires using the hyperreal numbers. That also seems far away from "basic number theory."
A Purely Algebraic Proof of the Fundamental Theorem of Algebra
Piotr Błaszczyk
(Submitted on 21 Apr 2015)
Proofs of the fundamental theorem of algebra can be divided up into three groups according to the techniques involved: proofs that rely on real or complex analysis, algebraic proofs, and topological proofs. Algebraic proofs make use of the fact that odd-degree real polynomials have real roots. This assumption, however, requires analytic methods, namely, the intermediate value theorem for real continuous functions. In this paper, we develop the idea of algebraic proof further towards a purely algebraic proof of the intermediate value theorem for real polynomials. In our proof, we neither use the notion of continuous function nor refer to any theorem of real and complex analysis. Instead, we apply techniques of modern algebra: we extend the field of real numbers to the non-Archimedean field of hyperreals via an ultraproduct construction and explore some relationships between the subring of limited hyperreals, its maximal ideal of infinitesimals, and real numbers.
Subjects: History and Overview (math.HO)
MSC classes: 08A40, 26E35
Cite as:
arXiv:1504.05609 [math.HO]
(or
arXiv:1504.05609v1 [math.HO] for this version)
