Timeline for Justifying a theory by a seemingly unrelated example
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 13, 2009 at 22:09 | comment | added | José Figueroa-O'Farrill | The topological proof follows from the computation of the fundamental group of the circle. Let $p = z^n + \dots \in \mathbb{C}[z]$ be a monic polynomial of degree $n>0$. If it has no roots on some disc $D$, then it defines a continuous map from $D$ to the punctured plane, which is null homotopic. However its restriction to the circle forming the boundary of the disc has degree $n$, being homotopic to $z \mapsto z^n$. Hence it must have a root, etc... | |
| Dec 13, 2009 at 21:47 | comment | added | Harrison Brown | There were "algebraic" proofs that predated Gauss, but they're really technical (they aim to be constructive!) and have wide gaps. I think they can be patched up, but not without some pretty heavy machinery (at least Galois theory). Are the topological and analytic proofs that different? The complex-analytic proof I know is basically an application of Cauchy's formula, which is a corollary of Stokes' theorem... Unfortunately I'm not that familiar with the topological proof, but Gauss' proof is more or less topological in nature. Fully analytical proofs I think require Cauchy. | |
| Dec 13, 2009 at 17:20 | comment | added | Thomas Bloom | You'd need to use some special fact about the real numbers though, since it is essentially a special property of $\mathbb{C}=\mathbb{R}+i\mathbb{R}. I think the impressive thing is that all that is needed is just the (really quite weak) MVT plus some algebra (elementary Sylow theorem and Galois theory stuff). | |
| Dec 13, 2009 at 17:16 | comment | added | Mariano Suárez-Álvarez | All the "algebraic" proofs I know of depend on things like the fact that odd-degree real polynomials have a real root (that is, on the mean value theorem). | |
| Dec 13, 2009 at 17:06 | history | made wiki | Post Made Community Wiki by Anton Geraschenko | ||
| Dec 13, 2009 at 16:38 | comment | added | José Figueroa-O'Farrill | There are analytic and algebraic proofs of the FTA, but not as simple as the topological one. Perhaps someone here knows the history better and can put the three proofs in chronological order? | |
| Dec 13, 2009 at 16:35 | comment | added | Mariano Suárez-Álvarez | Can the fundamental theorem of algebra be proven not using topology/trascendent methods? Mabe it is not made easier by topology but *possible :) | |
| Dec 13, 2009 at 15:06 | history | answered | José Figueroa-O'Farrill | CC BY-SA 2.5 |