Re: infinite vector spaces

drmwecker@xxxxxxxxx wrote:

With all due respect, Jose, first, I said "e.g.,", which means "for
example". I cited it precisely the same reason you did.

Second, I disagree that your mentioning polynomials having finitely
many zeroes somehow means "has nothing to do with the Fundamental
Theorem of Algebra". It most certainly does!

Please don't top-post. If you want to know what's that and why you
shouldn't do it, read

http://www.caliburn.nl/topposting.html

or

http://www.html-faq.com/etiquette/?toppost

On the other hand, the assertion

"a non-null polynomial has only a finite number of zeros" (*)

is valid for *every* field. Therefore, it has nothing to do with the
Fundamental Theorem of Algebra, which is a theorem concerning the field
of complex numbers. Yes, you are right, one can use the Fundamental
Theorem of Algebra in order to prove that a non-null polynomial with
coefficients in a subfield of the complex field has only a finite number
of zeros, but don't you think that by doing it you are using an
unnecessarily strong theorem in order to prove only a particular case of
the assertion (*)? Not to mention that several proofs of the Fundamental
Theorem of Algebra actually *assume* that the assertion (*) is true.
See, for instance, the proof based upon the Argument Principle which can
be found here:

http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra#Analytical_proofs

Best regards,

Jose Carlos Santos
.

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