1

I was reading through a section in my abstract algebra book (Nicholson) introducing field extensions and in particular Kronecker's Theorem. The author mentioned that there is no "purely algebraic proof of the fundamental theorem is known; that is, every proof involves a limiting process at some stage."

For those wondering, the theorem is as follows:

enter image description here

It's found on page 233 of Introduction to Abstract Algebra by Nicholson, 4th edition.

Now I'm asking why does a limiting process un-algebra something, and brings it into the world of analysis? I'm only starting real analysis, so I can't really find a good answer to this at the moment.

Bill Dubuque
  • 272,048
iwjueph94rgytbhr
  • 830
  • 4
    Generally algebra does not involve topology. – copper.hat Oct 23 '23 at 03:20
  • 1
    @copper.hat so are algebra and analysis human-made labels or something, is there not a precise way to define something to be a part of analysis or algebra? – iwjueph94rgytbhr Oct 23 '23 at 03:22
  • I'm not sure what you are asking, but generally a result that relies on topology would fall under the heading of analysis. – copper.hat Oct 23 '23 at 03:24
  • 1
    Some irony in the fact that the fundamental theorem of algebra has no purely algebraic proof. – copper.hat Oct 23 '23 at 03:28
  • Please state precisely the "Kronecker's Theorem" that you refer to (the name is not standard). – Bill Dubuque Oct 23 '23 at 05:10
  • @BillDubuque its the here – iwjueph94rgytbhr Oct 23 '23 at 22:41
  • But that document does not contain the cited remark about purely algebraic proofs. – Bill Dubuque Oct 23 '23 at 22:44
  • @BillDubuque My post is updated, unfortunately I can't provide the page itself because as far as I'm concerned there is no PDF version of this book. (also the link I provided was just a google search on the version of Kronecker's Theorem that I'm talking about) – iwjueph94rgytbhr Oct 24 '23 at 02:34
  • 2
    Likely the claim about "no purely agebraic proof..." does not refer to the theorem you quoted. Possibly it refers to some version of the fundamental theorem of algebra (which uses the quoted theorem). Please edit your question to clarify this since it will likely mislead many readers as it stands (since the quoted theorem uses no such limit process). – Bill Dubuque Oct 24 '23 at 02:41
  • Please read the textbook more carefully. It is not Theorem 4 but (as I surmised) the Fundamental Theorem of Algebra that he refers to as using a limiting process. I have added an excerpt in your answer. I recommend that you replace the image by the text, and unaccept that answer now that the question is clarified. – Bill Dubuque Oct 24 '23 at 02:52
  • @BillDubuque Oops, you're right I didn't read that right! – iwjueph94rgytbhr Oct 24 '23 at 03:37

2 Answers2

3

The purported distinctions between, "algebra" and "analysis" are almost entirely just conventions of traditional language. So it's basically "by definition" that "limits" are not "algebra".

Even though I've taught courses with (traditional) labels like "algebra" and "real analysis" and "complex analysis", in recent decades (!) I've quite deliberately talked about relevant mathematical ideas that are not within the traditional "purity" limitations of the supposed subject. :)

From one viewpoint, all the math that human beings can do is "algebra", since we cannot actually execute infinite processes. :) And so on.

paul garrett
  • 52,465
  • 1
    Dear Dr. Garrett, I like the pragmatic approach to these types of questions. It is always quite the delight when the methods of one field pop up in another when no such relationship was expected (either by our lack of vision or due to our conventions!). – Shrugs Oct 24 '23 at 04:58
  • 2
    @AHappy It's worth emphasis that "analytic" bias can sometimes prove a hindrance to algebraic reasoning, e.g. see this proof of Sylvester's determinant identity, which works in a universal determinant ring, and exploits that is is a domain, so we can cancel before evaluating (to remove an "apparent singularity" when the cancelled determinant is zero). Even some grad students and professors have had problems understanding how it avoids division by zero (presumably due to such analytic mental blocks). – Bill Dubuque Oct 30 '23 at 22:07
0

Imagine that you are a commercial truck driver. What is the most common thing that you do? You drive a truck, of course. This truck was built according to some engineering design, which, in turn, was based on some ideas and results from chemistry, physics and mathematics. Does it mean that truck driving is a part of engineering and/or chemistry, physics and mathematics? Of course, not. The main agenda of commercial truck driving is to move goods from one location to another using trucks, which is quite different from the agendas of engineering, chemistry, physics and mathematics. The same with algebra. Yes, some of the theorems which can be stated in purely algebraic terms (let's not argue what "algebraic" means here, it suffices to say that the exostence of solutions of polynomial equations belongs to algebra) are proven using tools from other fields of mathematics, and the Fundamental Theorem of Algebra happens to be one of the early examples of such theorems. (But "most" algebraic results are proven by algebraic methods.) What this example illustrates is the interconnectedness of mathematics. At some point, you will also discover how, say, Topology uses tools from Algebra, how Analysis uses tools from Topology, how Probability uses tools from Analysis, and how Algebra uses tools from Probability!

As you learn more math you will discover more examples of this interconnectedness phenomenon, to the point where you discover that no field of math is isolated from the result of mathematics, in the sense that it never uses tools (in the form of, say, theorems and definitions) from other fields. At the same time, as you learn more Abstract Algebra, you learn more about its agenda and how different it is from the agenda from, say, Topology or Analysis.

Moishe Kohan
  • 97,719