From: ullrich@math.okstate.edu (David C. Ullrich) Subject: Re: The number i (i^2 = -1) and FTA Date: Sat, 24 Nov 2001 19:02:10 GMT Newsgroups: sci.math Summary: Proofs of Fundamental Theorem of Algebra On Fri, 23 Nov 2001 16:39:35 +0100, Wilbert Dijkhof wrote: >Jon Miller wrote: > >> "G. A. Edgar" wrote: >> >> > > I think Gauss's original proof was algebraic rather than analytic. >> > > Certainly there are some algebraic proofs. >> > >> > Every proof uses some algebra and some analysis. You can use more >> > algebra to get away with less analysis, or vice versa. >> >> I seem to recall a proof that uses topology and avoids all the non-"trivial" >> algebra and analysis. It's in Munkres' book. > >Can you give us some hints about that, since I've never seen it. I don't know whether this is what's in Munkres, but you can give a proof that's fairly "topological" like so: Say P(z) = z^n + ... + a_1*z + a_0. Suppose that P has no zero in the plane. Now let C_r be the image under P of the circle of radius r about the origin. (Ie, C_r(theta) = P(r exp(i theta)) for 0 <= thetat <= 2Pi.) Consider the winding number of C_r about the origin. If r is large enough C_r must wind about the origin exactly n times (you can either wave your hands about how for large r the highest-order term is the most important or you can easily show that if r is large enough then C_r is homotopic in C\{0} to a curve that just wraps around a circle n times.) On the other hand the winding number of C_0 about the origin is 0. But the winding number is a continuous function of r, contradiction. > I've seen >how it can be proved that there's an infinite number of prime numbers >using topology, that's also a nice application :) > >Wilbert > > David C. Ullrich ============================================================================== From: Donu Arapura Subject: Fund. theorem of algebra via Lefschetz fixed point theorem Date: 19 Dec 2001 06:15:47 -0500 Newsgroups: sci.math.research A few years ago I found a proof of the fundamental theorem of algebra using the Lefschetz fixed point theorem. While it is certainly less elementary than the standard proofs, it may have some pedagogical value. I'll it sketch below. I suspect that similar proofs are known, although perhaps unpublished like this one. I'd be interested to hear of any. Sketch: It suffices to check that any complex monic polynomial has a root. Any such polynomial is the characteristic polynomial of some matrix (one can use the so called companion matrix). Thus one is reduced to showing that an nxn complex matrix A has an eigenvalue or equivalently an eigenvector. A may be assumed to be invertible, since otherwise 0 is an eigenvalue. Then A acts on complex projective space P = CP^{n-1} by sending the span of v to the span of Av. An eigevector corresponds to a fixed point under this action. Since the general linear group of C is connected, A can be connected by a path to the identity I (this can be done explicitly by writing A as a product of elementary matrices and deforming these to I in the obvious way). It follows that the A is homotopic to I, and therefore its Lefschetz number on P coincides with the Euler characteristic of P which is nonzero. Therefore, A has a fixed point on P. -- - Donu Arapura ----------------------------------------------------------------- Dept. of Math. | phone: (765) 494-1983 | Purdue University | email: dvb@math.purdue.edu | West Lafayette, IN 47907 | web: http://www.math.purdue.edu/~dvb | -----------------------------------------------------------------