ACTA UNIVERSITATIS APULENSIS
ON THE FUNDAMENTAL THEOREM OF ALGEBRA. PROOF BASED
ON TOPOLOGY
by
Severina Mera, Nicolae Mera
Abstract. In this paper we focus on the fundamental theorem of algebra, one of the most important results in mathematics. The work consists of two parts: the history of the fundamental theorem of algebra and the proof of this theorem, based on topology.
I. The history of the fundamental theorem of algebra.
The fundamental theorem of algebra, stating that any polynomial with a positive effective degree has at least a root in the complex number field, was first given by Jirarom in 1629 and Descartes in 1637; Of course their formulation was different from the one that is used nowadays.
The fundamental theorem of algebra was the focus of research for many algebraists during the last century. The impossibility of finding explicit formulae for the roots of polynomial equations of degree higher than 4 with real numbers as coefficients which can be now explained by the theory of Abel and Galois, gave rise to the natural question whether these equations do always have complex or real roots.
The first proof of the fundamental theorem of algebra was given by D’Alembert in 1746. Although the scientists of the 18th century did not find any drawbacks in this proof, it still seemed to be too analytical. The wider known rigorous proofs are nowadays based on the following methods and techniques
-the theorem of calculating the limit under the integral
-the symmetrical functions property and the Kronecker-Steinitz theorem -the Liouville theorem
-the variation of the variable of a function along a closed curve
All these proofs are using more or less the tools of mathematical analysis. They can be classified with respect to the tools that are used as :
1) Proofs based on an elementary mathematical tools. 2) Proofs based on topology and the thoery of functions.
The aim of this paper is to present the proof based on the properties of the topological degree.
Let
R
n be the n dimensional real euclidean space, G an open bounded subset ofR
n,G the closure of G and F(G) the boundary of G n
r
R
G
T
:
→
a transform of the closure G inR
n nz
z
1,...,
the coordinates of the vectorz
∈
R
n nT
Severina Mera, Nicolae Mera - ON THE FUNDAMENTAL THEOREM OF ALGEBRA. PROOF BASED ON TOPOLOGY
2
( )
21
1 2
=
∑
=
n
i i
z
z
the euclidean norm of z.J(T) the Jacobian of T , which exists when the partial
∂
T
j∂
z
i(
i
=
1
,
2
,...,
n
)
are well defined.The transform
T
:
G
−→
R
n is said to be differentiable if it is continuous on G and the partial derivatives∂
T
j∂
z
i(
i
=
1
,
2
,...,
n
)
exists and they are continuous on G. Let T :G→Rn be a differentiable transform andz
0∈
R
n, a vector such that( )
z
z
0T
≠
for z∈Fr( )
G .Let then Φ:
[
0,∞)
→R be a continuous function such that∫
Φ
( )
1 2⋅
...
⋅
=
1
n R
n
dz
dz
dz
z
and Φ( )
r =0 for r∈[
0,α
)
∪[
β
,∞)
where( )
( )
{
T z −z z∈Fr G}
< <
< min ,
0
α
β
0 .The degree of the differentiable transform T, with respect to the subset G and the vector
z
0, is defined as the numberd
(
T
,
G
,
z
0)
given by the multiple integral:(
)
=
∫
Φ
(
( )
−
) ( )
( )
Gn
dz
dz
z
T
J
z
z
T
Gz
T
d
,
0 0 1...
. In order for the degreed
(
T
,
G
,
z
0)
to be corectly defined it has to be independent on the functionΦ
. If we denote by(
0)
1T
,
G
,
z
d
Φ
andd
Φ
2(
T
,
G
,
z
0)
the degrees constructed with two different functions Φ1 and Φ2 thend
Φ
1(
T
,
G
,
z
0)
=d
Φ
2(
T
,
G
,
z
0)
.Let
T
1,
T
2:
G
→
R
n be two differentiable transforms andz
0∈
R
n a vector such thatT
z( )
z
≠
z
0 for z∈Fr( )
G andi
=
1
,...,
n
, If T1( ) ( )
z −T2 z <γ
for−
∈
G
z
, thend
(
T
1,
G
,
z
0) (
=
d
T
2,
G
,
z
0)
.Let
T
:
G
→
R
n be a continuous transformation andz
∈
R
n a vector such that( )
z
z
0T
≠
for z∈Fr( )
G .Let
T
m be a sequence of differentiable transforms withT
m:
G
→
R
n such that( )
z
z
0T
m=
for z∈Fr( )
G ,m
>
m
0 andT
m( )
z
→
T
( )
z
uniformly on G when∞ →
m . The degree of the continuous transform T with respect to the subset G is defined as
d
(
T
,
G
,
z
0)
lim
d
(
T
m,
G
,
z
0)
m→∞
Severina Mera, Nicolae Mera - ON THE FUNDAMENTAL THEOREM OF ALGEBRA. PROOF BASED ON TOPOLOGY
3
Theorem. Let
T
:
G
→
R
n be a continuous transform andz
0∈
R
n a vector such thatT
( )
z
≠
z
0 for z∈Fr( )
G . Ifd
(
T
,
G
,
z
0)
≠
0
then there is a z1∈G such that( )
z
1z
0T
=
.Corollary. Let
T
1,
T
2:
G
→
R
n be two continuous transforms andz
0∈
R
n. If for every z∈Fr( )
G the following inequality holds( ) ( )
2 1( )
01 z T z T z z
T − < − ,
then
(
T
1,
G
,
z
0) (
d
T
2,
G
,
z
0)
d
=
.II. Proof based on the properties of the topological degree.
Theorem. Let nbe a positive integer, f
( )
z a polynomial of degree n with complex coefficients, of the formf
( )
z
z
a
n 1z
n 1...
a
0n
+
+
+
=
−− with n≥1,
We assume
r
0>
0
, to be large enough such that forz
=
r
0 we have( )
02 0 2 1 0
1
r
a
r
...
a
a
z
f
z
n−
≤
n− n−+
n− n−+
+
,r
0n=
z
n.We consider the transforms T1,T2,
T
1:
G
−→
R
2,T
2:
G
−→
R
2, where( )
( ) ( )
{
2}
0 2 2 2 1 2 2 1
,
,z R z z r
z z
G= = ∈ + < , given by the equations,
( )
1( ) ( )
1 2 2 1 21
z
P
z
z
P
z
z
T
=
⋅
( )
1( ) ( )
1 2 2 1 21
z
Q
z
z
Q
z
z
T
=
⋅
whereP
1( )
z
1z
2 şiP
2( )
z
1z
2 are the real and the imaginary part of the expressionz
n, withz
=
z
1+
iz
2, andQ
1( )
z
1z
2 andQ
2( )
z
1z
2 the real and the imaginary part of the polynomial f( )
z .We have d
(
T2,G,0) (
=d T1,G,0)
>0. Then we obtain that there is(
z
z
)
G
z
1=
12⋅
22∈
such that T2( )
z1 =0, which means that the polynomial f( )
z has at least a complex rootz
1=
z
12+
z
22 which completes the proof.REFERENCES
[1]Arnold, B. H. “ A Topological proof of the fundamental theorem of algebra” American Math Monthly 56(1949), 465-466.
