Welcome to Gaia! ::

I've also started working on my thesis. Yippee skippee. Sorry I haven't been around, I've been going through a rough patch.

Some of you may have seen this proof before, but it was new to me, and I thought it spiffy enough to share a summary:

Proof of the fundamental theorem of algebra (stated as "every complex polynomial has a zero"):

Consider C as a plane.
Consider the usual mapping of R^2 to the sphere S^2 using stereographic projection.
Considered as a map, the complex polynomial P:C->C can be lifted to a smooth map f:S^2->S^2. Assume it's not the zero map.
A point in the domain of f is critical iff the corresponding complex point is a zero of P.
Since S^2 is compact and P is not identically 0, there are only a finite number of critical points.
By the compactness of S^2 all but a finite number of points in the codomain are regular values.
Hence the set of regular values is a connected set.
The map g:S^2 -> | f^{-1}(g)| is locally constant.
Since g is not identically 0 on the regular values, it is nowhere 0.
Hence f is onto.
Hence P is onto.
Hence P has a zero.

Win!

lol no worries. Things still suck - I have no idea what I'm doing with my life - but I'm pretty sure it's not math and it's not physics. At least I've narrowed things down somewhat.

Also, on your roots proof. You assume that P:C -> C=R^2 has a lift to f:S^2 -> S^2 that is smooth. Doesn't that require that lim_{|z| -> infty} P(z) exists? Otherwise your lift f will be discontinuous. I mean, it doesn't really hurt the proof; you can probably use some slick analytic argument to consider S^2 minus a small open ball around the pole of the stereographic projection, which would still be compact.

The part that I don't get, that I'd really like to get, is why zeroes of P translate into critical points of f. Shouldn't zeroes of P' be critical points?

Layra covered the smoothness of the lift, but I made two typos and you caught them... It's the zeros of P' that are the critical points! Not the zeros of P (zeros of degree two or higher are shared by a polynomial and its derivative--- but the polynomial P=3 has no zeros, and yet its derivative has an infinite number of them. Also the statement should read P' is not identically 0... of course-- I should also have stated earlier "assume P is not a constant polynomial". Instead of assume P is not identically zero.

What is this that you're working on?

0_o I have no idea how I'd ever get mine to be that short. There are times I hate our department, though. They've given us absolutely no guidelines on length, formatting, etc. and no help at all in terms of how we should choose an advisor, topic, second reader, etc.

The grad student I was working for last semester also did her undergrad here, and her thesis was 98 pages. I guess when you write about the Riemann hypothesis, you have a lot to cover.

What's your thesis on?

I'm not really worried about it, but I do kind of want to shave it down a bit.

Also, I really, really like dimensional analysis at the moment, considering that I'd forgotten basically every formula taught so far in this physics class but managed to reproduce them using just dimensional analysis (up to dimensionless constants).