Proof of ARGAND's Inequality. The decisive role in the proof is played by the following

ARGAND

3. Proof of ARGAND's Inequality. The decisive role in the proof is played by the following

Lemma. Let k be a natural number, not zero, and let

h:=l+bZk+Zkg

with

bECX, 9EC[Z], g(O)=O.

Then there is a ii C such that Ih(u)I < 1.

Proof. We choose a kth root d E C, of—i/b, so that = ^—1 (proposition 2' of the introduction). For all real I with 0 < I < 1, we then have

Ih(dt)I ^— + ^{= 1—} +

Since g, being a polynomial, is continuous at 0 (proposition 0 of the intro- duction), and since g(0) = ^0,

there exists a 6, with 0 < 6 < 1, such that

< for all I satisfying the inequality 0 < I < 6. For every such I, it then follows that Ih(di)I < 1 —^1k

+

^{< 1.}

The reader will notice that, apart from g(0) =0, the only property of

g: C C which has been used, is that of continuity at the origin. The lemma therefore holds for all such functions. The argument shows that h assumes values less than 1 in an arbitrurilg, small neighborlzood of the origin.

ARGAND'S inequality now quickly follows: a nonconstant 1(Z) implies that 1(Z) := f(c + Z)/f(c) E C[Z] is not constant. Now

/(Z)= 1+bkZk

with bk 0,

1 <k

^n•

Writing g(Z) := bk+IZ + + we have /

= 1

+ +

^Zkg with g(0) = 0. By the Lemma there exists therefore an u E C, such that

I < 1. For c' := c + u, we then have

If(c')I = Ih(u)i If(c)I <If(c)I. 0 In function theory ARGAND'S inequality is a special case of the gen- eral "open mapping theorem" which asserts that nonconstant holomorphic functions always map open sets on to open sets. (See J. Conway, Functions of One Compkz Variable, Springer-Verlag, 1978, p. 95.)

4. Variant of the Proof. We describe here a variant of the proof of the

Fundamental theorem in which the existence of kth roots, with k > 2, is assumed only for positive real numbers, and their existence for arbitrary complex numbers is proved as a consequence. We use induction on the

degree of the polynomial f, the initial step in the induction being clear.

Since the polynomial / defined in the previous paragraph has the same degree as f and since the truth of of Lemma 3 for all polynomials h of

degree < n follows from the truth of the Fundamental theorem for all

polynomials of degree < n (via the ARGAND inequality) it suffices to show that:

If the Fundamental theorem holds for all polynomials of degree < n, n 2, then Lemma 3 holds for cli polynomials h of degree n.

Let h be any of the admissible polynomials of degree n in Lemma 3. We distinguish three cases:

(1) k < n. Then by hypothesis the Fundamental theorem holds for all polynomials — a,

a E C; all a

C therefore have kth roots and the lemma can be proved as in 3.

(2) k = n, with n even. Then h = ¹ + bz" with b 0. Choose a square

root 'i of —1/b and let u be a k/2th root of

(which is allowable since k/2 < n); it then follows that h(u) = 0 < 1.

(3) k =

^n,

with n odd. Again h =

¹

+

with b 0. One can then find a u C satisfying Ii + < 1 in the following amusing way: for

c := —IbI/b E S' there is a w E {1, —1,i, —i) such that Ic _{— wI <} 1 (see Exercise 3.3.4). As n is odd, the set { 1,—i, 1, —i) is mapped onto itself by the transformation z i—. z", and there is therefore a v E C such that

=

w. For u :=v/ C we have ^. = wand hence

=

—w/c.

Since id = 1 it follows that

1. 0

The first inductive proof of this kind was given in 1941 by J .E. LITTLE- WOOD: "Mathematical notes (14): every polynomial has a root." J. Lond.

Math. Soc., 16, 95-98. An even simpler proof was given in 1956 by T. Es- TERMANN "On the fundamental theorem of algebra," J. Lond. Math. Soc.,

31, 238—240.

5. Constructive Proofs of the Fundamental Theorem. The AROAND- CAUCHY proof is purely an existence proof and is non-constructive. As early as 1859 WEIERSTRASS in his note "Neuer Beweis des Fundamentalsatzes der Algebra" (Math. Werke 1, 247—256) had made the following start to- wards a constructive proof: given a polynomial 1(Z), a number z0 := c E C

ischosen arbitrarily and the sequence := ^—

I

defined recur- sively. WEIERSTRASS says (p. 247) "...

it

can be shown that when ii is

increased indefinitely, z,, under certain conditions, tends to a limit z satis- fying the equation 1(z) = 0." More than 30 years later (1891, Math. Werke 3, 25 1—269) WEIERSTRASS once again discusses in detail the problem of a constructive proof by asking the following question:

"Is it possiblefor any given polynomial f E C[Z], to produce a sequence of complex numbers by an effectively defined procedure, so that

is sufficiently small in relation to that it converges to a zero of f?" H. KNESER in 1940 in his paper entitled "Der Fundamentalsatz der Algebra und der Intuitionismus," Math. Z., 46, 287—302, defined such a process which yields a constructive variant of the AROAND—CAUCHY proof and which also satisfies the criticisms of the intuitionists. M. KNEsER in 1981 further simplified his father's process in a paper entitled "Ergänzung zu einer Arbeit von Heilmuth KNESER über den Fundamentalsatz der Al- gebra," Math. Z., 177, 285—287.

In 1979 HIRSCH and SMALE described a "sure fire algorithm" which produces, for any nonconstant polynomial 1(Z) CEZI and any arbitrary initial point c E C asequence z,1, with Zo = c, which converges to a zero of 1. More precisely it is shown that:

(*)

n0,1,2,...

with a positive real constant K < 1, depending only on the degree off, not on I itself. For details, see the article "On algorithms for solving 1(z) = 0"

in Comm. Pure Appi. Math., 32, 281—312 and in particular pp. 303 ci seq.

The inequality (*), and with it a "sure fire algorithm" is already to be

found in KNESER, bc. cii., p. 292, formula (6), except that, to satisfy the demands of the intuitionists, If(c)I is replaced by Max(1, If(c)I).

§3. APPLICATION OF THE FUNDAMENTAL THEOREM

The existence of at least one zero for every nonconstant complex polyno- mial already implies that complex polynomials decompose into linear and that real polynomials decompose into linear and quadratic factors. These consequences of the Fundamental theorem are completely elementary, and are a result of the simple fact that a polynomial with a zero at c always has the factor z —c.

1. Factorization Lemma. If c

C is a zero ofthe polynomial f E C[Z]

of degree n, then there is just one polynomial g C[Z] of degree n—1, such

that f(Z) =

(Z^—c)g(Z).

Proof. Let f =

^. 0. Since Z"—c" = (Z—c)q,.(Z) with :=

+ Z"2c + ..

+ it follows that

1(Z) =

1(Z)

-

^{f(c) =}

-

^{c")= (Z}

-

^c)g(Z),

where

g(Z) :=

It is clear that f is of degree n — ^1: since g(z) = (z — z c, g is

uniquely determined by f and c.

0

The factorization lemma holds for alt commutative rings, provided that one gives up the uniqueness of g. By induction on ii we at once obtain the Corollary. A polynomial f E C[ZJ of degree n has at most n zeros.

2. Factorization of Complex Polynomials. Every complex polynomial I E C[Z] of degree n 1 is, disregarding the order of the factors, uniquely representable in the form

(1) 1(Z) = a(Z — ^{— .} . (Z ^—

where a E CX; r E N, C1, .. ^{., C,.} E C are ^distinct from one another, and

flu

...

^{, t2,.} EN\ {O}

fli+

+ + flr = fl

Proof. We use induction on n, the case n = I being clearly true. Suppose

n >

^1. By the Fundamental theorem of algebra there exists a C for which f vanishes. By lemma!, 1(Z) = (Z—ci)g(Z), where g(Z) E C[ZJ is

of degree n — 1. By the inductive hypothesis there is a unique factorization

g(Z) = o(Z — ^{— .}... . (Z —

with

distinct from one another, and a E CX. Consequently (1) holds.

0

The theorem just proved is often stated in the form:

Every complex polynomial of the nih degree has precisely n zeros where each of the zeros c, is counted according to its multiplicity