The following well-known theorem will be proved first, and then the so-called Fundamental Theorem of Algebra will be deduced from it.
Preliminaries: 1. The angular variation is defined for , with by the equation
2. For a continuous curve
the angular variation along is defined to be
4. For a constant curve we have . To see this, compute with .
5. For the curve , we have . Compute with .
6. For a continuous family of closed, continuous curves we have
is uniformly continuous. Choose
in such a way that for all
holds. The function
Proof of the theorem:Choose a basis
, for and consider the function
Choose in two linear independent vectors and , and
consider the family of curves
Proof:Consider a chain of maximal length of linear subspaces in V, with and Then and induces on each quotient space a linear transformation, say such that there is no subspace in with and So, by the theorem we have Then and the subspace has the property of the corollary. A system such that is a basis of for which the matrix of is uppertriangular.
be a polynomial of degree .
Consider the linear transformation
on the standard basis
of as follows:
Remark. The Fundamental Theorem of Algebra was proved by C. Gauss [G]. As is done in the textbooks on Linear Algebra, one can use it also to proof directly the existence of eigenvectors for endomorphism of finite dimensional vector spaces over . A first appearance of eigenvectors for symetric matrices is in the work of Cauchy on inertials elipsoids (see the textbook of P.M. Cohn [C]). The Jordan Normal Form Theorem is formulated and proved by C. Jordan [J] for general fields . If any polynomial of positiv degree factors in in polynomials of degree one, the Normal Form Theorem proofs the existence of invariant flags for endomorphisms of finite dimensional vector spaces over the field .
[C] P.M. Cohn, Algebra, Wiley, New York, 1982.
[J] Camille Jordan, Traité des substitutions et équations algébriques, Gauthiers-Villars, Paris, 1957.
[G] C. Gauss, .