Some Riemannian geometric proofs of the fundamental theorem of algebra
If a (non-constant) polynomial has no zero, then a certain Riemannian metric is constructed on the two dimensional sphere. Several geometric arguments are then shown to contradict this fact. M.S.C. 2010: 12E05, 53C20.
Yet another application of the Gauss-Bonnet Theorem for the sphere
where K and dAg denote the Gaussian curvature and the area element of g, respectively. In particular, this implies the existence of at least an elliptic point of (S, g). In this short note we prove the Fundamental Theorem of Algebra as a consequence of this fact. As far as the authors know, this proof is new. Moreover, it does have the advantage of being directly based on the work of Gauss, who was the first mathematician giving an universally accepted proof of the Fundamental Theorem of Algebra (see [2], [3]). Hence our argument can also have some historical and pedagogical interest. As it is well known, the stereographic projection allows to identify the sphere S with the one-point compactification Ĉ = C ∪ {∞} of the complex plane C. Let us take p(z) = a0 + a1z+ · · ·+ anz n with a0 an 6= 0 and let us assume that p(z) 6= 0 for all z ∈ C. We set p∗(z) = an + an−1z + · · ·a0z , that is, the i-th coefficient of p∗(z) is the (n− i)-th coefficient of p(z). Clearly, p∗(z) = z n p(1/z) for all z ∈ C \ {0}. Put
Philosophy in a new key : a study in the symbolism of reason, rite, and art
A Prefatory Notes to the Third Edition Preface to the Edition of 1951 Chapter 1: The New Key Chapter 2: Symbolic Transformations Chapter 3: The Logic of Signs and Symbols Chapter 4: Discursive Forms and Presentational Forms Chapter 5: Language Chapter 6: Life-Symbols: The Roots of Sacrament Chapter 7: Life-Symbols: The Roots of Myth Chapter 8: On Significance in Music Chapter 9: The Genesis of Artistic Import Chapter 10: The Fabric of Meaning