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:
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.
Best Answer
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.