Proof of the Fundamental Theorem of Algebra
Author(s): S. Wolfenstein
Source: The American Mathematical Monthly, Vol. 74, No. 7 (Aug. - Sep., 1967), pp. 853-854
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2315822 .
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1967]
853
CLASSROOM NOTES
PROOF OF THE FUNDAMENTAL THEOREM OF ALGEBRA
de Paris
S. WOLFENSTEIN,
Universite
The so-called Fundamental Theorem of Algebra states that every nonconstant polynomial with complex coefficientshas at least one complex zero; or
alternatively,that if p is any nonconstant polynomial functionof a complex
variable, then the range of p contains every complex number.
The purpose of the present note is to sketch a proofsimpler than appears
currentlyavailable and in any case employing no mathematical tools more
sophisticated than are already required for elementarycalculus. The proof is
based upon the convergenceof an iterative procedure,a notion which should
be familiar to beginningstudents (Newton's method, Picard's method). Prerequisitesare as follows:
(A) A nonzeropolynomialhas only a finitenumberof zeros.
(B) If f: Rm >Rnis a continuous functionsuch that If(x)I goes to infinity
with lxI (in particular iff is a nonconstantpolynomial functionof a complex
variable), then the range off is closed. (This followsat once fromWeierstrass's
Theorem that everybounded sequence has a convergentsubsequence.)
(C) For n> 1, the complement of a finitesubset of Rn is connected. In
addition only the most elementarypropertiesof complex numbersare needed.
LEMMA.Let p be a nonconstantpolynomialfunctionof a complexvariable,S
therangeofp and T= {p(z): p'(z) =01. Then S-T is open.
small the equation
Proof. We firstshow that if Ic| is sufficiently
(1)
W+
n
bmWmc
m=2
has a root that can be approximatedby iteration.Let wo= 0,
n
>,
Wk+1 = c -
m
bmWkX
m=2
clearly if the sequence wk convergesat all it convergesto a root of (1). Suppose
now that |c is small; specifically,let a be any positive number less than 1,
p-=min(Oa/[E =2 mIbmj], 1) and I|c <P- En=2 |bin pm(this is a positive
inductivecalculation shows that, forall k, Iwkf <p.
number). A straightforward
Further, for k> O,
Wk+1
-
Wk -
WkI |
<
|
Wk-1I
n
Zbm
E
rn=2
n
-
< at I Wk -
Wk_l
IE
m=2
Wk-1
m
Em8W
J-I
8=1
m I bmI pm-l,
by theforegoing,
854
CLASSROOM
[September
NOTES
so that, forc as stated, Wk does convergeto a root of (1).
Now let p be any nonconstant polynomial, zo any complex number such
thatp'(zo) O. With the substitutionsz=zo+w, a==p(zo)+p'(zo)c, the result
just obtained shows that the equation p(z)=a has a root for all values of a
close enough to p(zo). In otherwords,ifzo is not a zero of p', p(z0) is an interior
point of S. This proves the Lemma.
THEOREM.
Withthenotationof thelemma,S is theset ofall complexnumbers.
Proof. The complementof S is open (B), and S- T is open by the lemma.
But as T is finite(A) its complementcannot consist of two disjoint nonempty
open sets (C). Hence the complementof S must be empty.
AN EASY PROOF OF THE FUNDAMENTAL THEOREM OF ALGEBRA
CHARLES
FEFFERMAN,
University
ofMaryland
+an2Znbe a complex polynomial. Then P
THEOREM. Let P(z) ==ao+alz+
has a zero.
Proof.We shall show firstthat IP(z) I attains a minimumas z varies over the
entire complex plane, and next that if IP(zo) is the minimum of |P(z) |,
then P(zo) =0.
Since JP(z)j
so large that
|zfnIa+an
+ao/znJ
l/z+
(1)
(z20),
wecanfindan
I aolI
IP(z)|I
(IzI
M>O
>M).
Now, the continuous function|P(z)j attains a minimumas z varies over the
compact disc { z Izj
I M}. Suppose, then,that
(2)
|
P(zo) I
Z
I|
_ I P(Z) I
?<M).
In particular, P(zo) | P(0) = Iao I so that, by (1), P(zo) ?<-| P(z)| (I zI >M)
Comparing with (2), we have
| P(zo)
(3)
I ' IP(Z) I
Since P(z) =P((z-zo)+zo),
(all complexz).
we can write P(z)
so that forsome complex polynomial Q,
P(z)
(4)
Q(z
-
as a sum of powers of z-zo,
Zo).
By (3) and (4),
(5)
| Q(O)|
Q(z)
(all complexz).
We shall show that Q(O) =0. This will establish the theorem since, by (4),
P(zo) ==Q(O).
Let j be the smallest nonzero exponent for which zi has a nonzero coefficient in Q. Then we can write Q(z) =co+cjzi+
+cnzn (cj1-zO). Factoring