**Norbert A'Campo**

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

For with , and the angular variations obey the following additive rule:

**2.** For a continuous curve
the angular variation along is defined to be

where , is chosen in such a way that for all with the inequality holds. The angular variation does not depend on the actual choice of since for two such choices and it follows from the additive rule: The angular variation satisfies:

**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

**Proof:**The function
is uniformly continuous. Choose
in such a way that for all
with
the inequality
holds. The function

is continuous, with values in by 3, and therefore constant. Hence

**Proof of the theorem:**Choose a basis
,
, for and consider the function

where is the linear transformation given by

We have for . For any with and the subspace satisfies: Therefore it suffices to prove the existence of a vector with and

Fig. 1.

Choose in two linear independent vectors and , and
consider the family of curves
given by:

We claim: there exists such that , and the vector satisfies and . We prove the claim by contradiction. Assume for all . Observe

We obtain: using the preliminaries 5, 6 and 4, which contradicts the hypothesis .

**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.

**Proof:**Let
be a polynomial of degree .
Consider the linear transformation
acting
on the standard basis
of as follows:

Clearly and , . We conclude that . Thus for an eigenvector , with eigenvalue , we get , and hence .

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, .