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PRAYAS
Students’ Journal
of Physics
An algebraic topological proof of the fundamental theorem of algebra
Gouri Shankar Seal (3rd year Integrated M.Sc student)
Indian Institute of Science Education and Research, Block HC-VII, Sector-III, Kolkata-700106
Abstract. Several proofs of the fundamental theorem of algebra, using purely algebraic and complex analytic
(via Liouville’s theorem) methods are well known. The algebraic topological version of the proof is an elegant
application of elements of homotopy theory. In this article, the requisite concepts are introduced first and then
the proof has been presented.
Communicated by: P.K. Panigrahi
1. HOMOTOPY OF PATHS
Definition 1.1:(path) Let X be a topological space. A path in X joining x0 and x1 is a continuous
function f : [0, 1] → X such that f (0) = x0 and f (1) = x1 .
Definition 1.2:(path homotopy) Let X be a topological space. Let I be the unit interval [0, 1]. A
homotopy of paths in X is a family of paths ft : I → X such that
(i)ft (0) = x0 , ft (1) = x1 , are independent of t
(ii)F : I × I → X defined by F (s, t) = ft (s) is continuous; 0 ≤ s ≤ 1; 0 ≤ t ≤ 1, such that
F (s, 0) = f0 (s) and F (s, 1) = f1 (s).
Fig 1: Homotopy as a continuous deformation of paths
For visualization one can view s as a space parameter and t as a time parameter.
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Gouri Shankar Seal
Example 1.1:(linear homotopy) Any two paths in Rn having same initial and final points are homotopic via the homotopy ft (s) = (1 − t)f0 (s) + tf1 (s).
We can see from the above definition that the homotopy of paths is an equivalence relation in the
set of paths from I → X. The equivalence class of paths in X under the equivalence relation of
homotopy is denoted by [f ].
Definition 1.3:(composition of paths) Let f ,g be paths in a topological space X such that f (1) =
g(0). We define the product path
(
f (2s)
0 ≤ s ≤ 21 ;
f.g =
g(2s − 1) 12 ≤ s ≤ 1
Fig 2: Composition of two paths
It turns out that the equivalence path class defined above has all the requisite properties of a group
except associativity (i.e., (f.g).h 6= f.(g.h)): this is called a groupoid structure. In order to get the
group structure, we must look at closed paths or loops in X.
2. THE FUNDAMENTAL GROUP
Definition 2.1:(loops) A loop based at a point x0 on a topological space X is a continous function
f : I → X such that f (0) = f (1) = x0 .
Theorem 2.1: Let X be a topological space. The set of all homotopy classes of loops based at x0 is
a group under the product [f ].[g] = [f.g], where f and g are loops based at x0 . This is denoted by
π1 (X, x0 )(also known as the first Homotopy group of X).
Fig 3: A loop is a closed path
Proof: We take the identity element of the group to be the constant loop based at x0 denoted by
C(x0 ). Inverse of [f ] is [f −1 ] where f −1 (t) = f (1 − t); 0 ≤ t ≤ 1.
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Prayas Vol. 3, No. 4, July - August 2008
An algebraic topological proof . . . fundamental theorem of algebra
Definition 2.2:(path connected spaces) A topological space where any two points can be joined by
a path.
Proposition 2.1: If X is a path connected topological space then for all points x0 ∈ X the groups
π1 (X, x0 ) are isomorphic
Proof: The map βh : π1 (X, x1 ) → π1 (X, x0 ) defined by βh [f ] = [hf h−1 ] is an isomorphism. If ft
is a homotopy of loops based at x1 then [hft h−1 ] is a homotopy of loops based at x0 .
Fig 4: In path connected spaces choice of the base point is arbitrary.
Definition 2.3:(simply connected space) A topological space which is path connected and has trivial fundamental group (π1 (X) = 0). This means any loop based at x0 in X is homotopic to the
constant loop C(x0 ) based at x0 ! As an example of a simply connected space try to visualize the
usual two dimensional Euclidean plane (R2 ) and a rubber band loop based at any point in the space.
One can continuously deform the rubber band and shrink it to the base point demonstrating that
there is a unique homotopy class of loops.
Fig 5: Simply connected spaces have trivial fundamental group
3. THE FUNDAMENTAL THEOREM OF ALGEBRA
We now state an important result that the fundamental group of the unit circle S 1 based in R2 is
isomorphic to Z.
Theorem 3.1: φ : Z → S 1 sending an integer n to the homotopy class of the loop ωn (s) =
(cos(2πns), sin(2πns)); 0 ≤ s ≤ 1 based at (1, 0) is an isomorphism.
Prayas Vol. 3, No. 4, July - August 2008
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Gouri Shankar Seal
We outline the idea of the proof. Choose the base point at (1,0). Let f : [0, 1] → S 1 be a loop. When
n = 1, f completes a turn and returns to (1,0). Therefore the total number of times it wounds around
the circle is an integer. This number (called the winding number) is unchanged by the deformation of
the loop and under a continuous deformation of the loop remains a constant. Hence, every homotopy
class has a unique winding number and φ is an isomorphism.
We are now in a position to state and prove the fundamental theorem of algebra:
Theorem 3.2: Every nonconstant polynomial with coefficients in C has a root in C
Proof: Assume p(z) = z n + a1 z n−1 + . . . + an . If p(z) has no roots in C then define:
p(r exp(2πis))/p(r)
; r ≥ 0 a loop in the unit circle S 1 based at (1,0). As r varies starting
|p(r exp(2πis)/p(r)|
from 0, fr is a homotopy of loops based at (1,0). Since f0 is the trivial loop [fr ] ∈ π1 (S 1 ) is zero
for all r.
Fix r > (|a1 | + |a2 | + . . . + |an |) and r ≥ 1. Then for |z| = r, we have |z n | = rn = r.rn−1 >
(|a1 | + . . . + |an |)(|z n−1 |) ≥ |(a1 z n−1 + . . . + an )|, or|z n | > |(a1 z n−1 + . . . + an )|.
fr (s) =
Now, define pt (z) = z n + t(a1 z n−1 + . . . + an ). If possible let pt (z) has a root in |z| = r .
Setting pt (z) = 0 we get z n = −t.(a1 z n−1 + . . . + an ) which gives |t||a1 z n−1 + . . . + an | >
|a1 z n−1 + . . . + an | which gives |t| > 1, a contradiction. Hence pt (z) has no roots on the circle
|z| = r, when 0 ≤ t ≤ 1.
Now, replacing p by pt in the formula for fr and letting t go from 0 to 1, we obtain a map
pt (r exp(2πis))/pt (r)
. It is easy to see that F0 (s) = ωn (s) and F1 (s) = fr (s), clearly
Ft (s) =
|pt (r exp(2πis)/pt (r)|
Ft (s) defines a homotopy from the loop fr to ωn (s) = exp(2πins) i.e., [ωn ] = [fr ]. But by our
construction we know [fr ] = 0 then [ωn ] = 0 which implies n=0. So the only polynomials in C
without roots in C are the constant polynomials.
Acknowledgement
The author would like to thank Prof.Prasanta K. Panigrahi for his constant encouragement. He
would also like to thank Mr.Gaurav Tripathi for his help.
References
[1] Allen Hatcher: Algebraic Topology, 2001, Cambridge University Press, Cambridge, U.K.
[2] William S Massey: Algebraic Topology: An Introduction, 1990, Springer, New-York, U.S.A.
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