Re: [math-fun] The Axiom of Choice for roots of z^2 + 1
Kerry wrote: << [I wrote:] << There is absolutely no mathematical way to distinguish between [the two complex square roots of -1]. Perhaps they should only be referred to as a pair, and never one at a time?
Ok, I'll show my ignorance here. How can we distinguish between 1 and -1 and not be able to distinguish between i and -i? Isn't the simple fact that they're distinct numbers (or distinct points in the complex plane) enough? What am I missing?
Well, 1 and -1 have different algebraic properties. E.g., 1 is the unique complex number whose product with any other number z is z. -1 is the unique complex number that's unequal to 1 but whose square is 1. On the other hand, there is no distinguishing property that holds for one of i,-i but not the other. This follows from the fact that there exists a field isomorphism f:C -> C interchanging i and -i. (The simplest example is complex conjugation: f(x+iy) = x-iy.) --Dan
Well, 1 and -1 have different algebraic properties. E.g., 1 is the unique complex number whose product with any other number z is z. -1 is the unique complex number that's unequal to 1 but whose square is 1.
Thanks Dan and Joshua. That's what I get for being ignorant--I learn stuff. :-) Kerry
A remark in one of the papers of Abhyankar says something similar to what Dan has pointed out. "Originally the equation Y^2 + 1 = 0 had no solution. Then the solutions `i' and `-i' were created. But there is absolutely no way to tell who is `i' and who is `-i'. That is Galois theory." Needless to say: this was an introduction in one of his papers on Galois theory. There is also an interesting footnote on the same page. The link to the paper is here: http://arxiv.org/abs/math.NT/9207210 Best regards, Shripad. On Nov 24, 2007 6:22 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Kerry wrote:
<< [I wrote:]
<< There is absolutely no mathematical way to distinguish between [the two complex square roots of -1]. Perhaps they should only be referred to as a pair, and never one at a time?
Ok, I'll show my ignorance here. How can we distinguish between 1 and -1 and not be able to distinguish between i and -i? Isn't the simple fact that they're distinct numbers (or distinct points in the complex plane) enough? What am I missing?
Well, 1 and -1 have different algebraic properties. E.g., 1 is the unique complex number whose product with any other number z is z. -1 is the unique complex number that's unequal to 1 but whose square is 1.
On the other hand, there is no distinguishing property that holds for one of i,-i but not the other. This follows from the fact that there exists a field isomorphism f:C -> C interchanging i and -i. (The simplest example is complex conjugation: f(x+iy) = x-iy.)
--Dan
participants (3)
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Dan Asimov -
Kerry Mitchell -
Shripad M. Garge