Monday, September 15, 2014

Mathbox update: Fundamental Theorem of Algebra

The main theorems in this short update are Van der Waerden's theorem (already discussed in a previous post) and the Fundamental Theorem of Algebra. This last one is one of the metamath 100, and the proof was surprisingly straightforward. The hardest part was the first step, to show that a closed disk in $\Bbb C$ is compact via the Heine-Borel theorem; then it is just a matter of verifying some explicit bounds. By the way, this proof is based on the first "topological proof" on Wikipedia.
• vdw: Van der Waerden's theorem: for any $k$ and finite set $R$ of colors, there is an $N$ such that any $R$-coloring of $\{1\dots N\}$ contains a length-$k$ monochromatic arithmetic progression. See also the previous post I wrote on this theorem.
• vdwnn: Infinitary Van der Waerden's theorem: for any finite coloring of $\Bbb N$, there is a color that contains arbitrarily long arithmetic progressions.
• txcmp, txcmpb: The topological product of two nonempty spaces is compact iff both factors are compact.
• hmeores: The restriction of a homeomorphism of topological spaces is a homeomorphism on the induced subspace topologies.
• cnheibor: Heine-Borel theorem for the complex numbers: A subset of $\Bbb C$ is compact iff it is closed and bounded. Note that we already have the more general heibor (a metric space is compact iff it is complete and totally bounded), but this direct proof avoids the axiom of countable choice.
• dgrco: The degree of a composition of polynomials is the product of the degrees.
• fta: The Fundamental Theorem of Algebra: Every non-constant polynomial has a root.