Zum Fundamentalsatz der Algebra. (On the fundamental theorem of algebra) (Q1080479)

The author provides a proof of the fundamental theorem of algebra based on a generalized Sperner lemma: Let a triangle \(\Delta_ 0\) be decomposed into triangles \(\Delta\) and let all the corners in the partition be marked 1 or 2 or 3. With regard to a fixed positive sense of rotation let all \(\Delta\) be signed: \(\alpha(\Delta)=+1\) or -1 if \(\Delta\) has its corners marked 1,2,3 or 1,3,2, and \(\alpha(\Delta)=0\) if two corners of \(\Delta\) carry the same number. Let \(\alpha(1,2)=- \alpha(2,1)=1\) and \(\alpha(s)=0\) for the other sides s. Then \(\sum_{\Delta \subset \Delta_ 0}\alpha (\Delta)=\sum_{s\subset Rd \Delta_ 0}\alpha (s)\). \{This is readily counted in a given figure; a real proof, however, is not provided. The subsequent proof of the theorem is the most complicated one the reviewer has seen so far.\}