Information Technology Reference
In-Depth Information
argument, and so we eventually reach a polynomial of degree 1, having exactly one
solution.
In conclusion, the essential part of the fundamental theorem is the proof of the
existence of at least one solution for any algebraic equation
P
(
)=
∈
C
.
After some partial proofs of Euler, a first complete attempt of proof of the fun-
damental theorem is that given by d'Alembert in 1746, assuming a lemma, which
was stated without a rigorous proof (Argand proved it in 1806 [166]). During his
lifetime, Carl Friedrich Gauss offered at least four different proofs of this theorem,
covering the timespan of his entire adult life. His first proof was published in his doc-
toral dissertation in 1799, at age of 22. In 1849, just a few years before his death,
Gauss gave a fourth proof that bore similarity to his first one. These proofs (the
first and the fourth one) were grounded on subtle and deep properties of continuous
functions and topological concepts, which at that time were missing of any rigorous
clarification (the masterpieces of Dedekind and Weierstrass about reals and conti-
nuity were completed after 1870, and topological properties of curves, implicitly
assumed in the first and fourth proofs of Gauss, reach a completely rigorous math-
ematical elaboration only in the 20th century). Quoting from [192], Gauss claims
that: “It seems to be well demonstrated that an algebraic curve neither ends abruptly
(as it happens in the transcendental curve
y
z
0, with
z
log
x
), nor loses itself after an infi-
nite number of windings in a point (like a logarithmic spiral)”.
The informal proof that we outline uses arguments different from those of Gauss
(see [171] for a topological proof). Let us assume the concept of transformation of
the complex plane; for any polynomial
P
, any point
z
of the complex plane trans-
forms into point
P
=
1
/
(
z
)
of the same plane. Let us fix a polynomial
P
(
z
)
(with complex
2 and where the monomial
z
k
has coefficient 1 (this
does not imply any loss of generality). Let us transform the points of a disk
D
of the
complex plane having as boundary a circle
C
, the center in the origin, and radius
R
.
Of course,
P
coefficients) having degree
k
>
belongs to
D
(
P
0, otherwise the proof would be concluded,
being the origin a solution of the equation). The set of points that are
P
-images of
the points in
D
form a region
D
, with boundary
C
. The polynomial
P
transforms
interior points of
D
into interior points of
D
and points on the boundary
C
into
points on the boundary
C
. This intuitive property has a rigorous explanation, based
on analytical aspects related to the “open mapping theorem”. Let us define the ra-
dius
R
of
D
with respect to
P
(
0
)
(
0
)
=
and
C
.
For large enough and increasing values of the radius
R
of
C
, the monomial
z
k
of
P
becomes dominant (on the points of
C
), thus the corresponding radius
R
of
D
(with
respect to
P
(
0
)
as the minimum distance between
P
(
0
)
) increases consequently. Therefore, for a sufficiently large radius
R
of
C
the radius
R
of
D
, is greater than the distance between
P
(
0
)
and the origin. In
conclusion, when the radius
R
of
C
reaches this value, the origin belongs to
D
,so
that a point
z
0
in
D
has to exist which is transformed into 0, that is,
P
(
0
)
0.
The representation of complex numbers in the plane is an example of geometri-
cal representation of pairs of numbers as points in a two-dimensional space. This
idea easily generalizes to triples and, in general, to any sequence of
k
components.
In other words, if
(
z
0
)=
2
3
R
corresponds to the plane and
R
corresponds to the three-
k
dimensional space, then
R
can be seen as a
k
-dimensional space, where basic
