Proofs of the Fundamental Theorem of Algebra

In his first proof of the Fundamental Theorem of Algebra, Gauss deliberately avoided using imaginaries. When formulated for a polynomial with real coefficients, the theorem states that every such polynomial can be represented as a product of first and second degree terms. Second degree factors correspond to pairs of conjugate complex roots. Over the field of complex numbers a more elegant formulation is possible: every polynomial is a product of first degree terms. The latter formulation is not only more elegant, it's also more revealing. Staying in the realm of real numbers it's hard to explain why and wherefrom quadratic terms appear. Complex numbers provide an immediate explanation.

Complex numbers indeed proved to be a natural setting for the theorem. But the realization of course did not come immediately. The father of the modern complex analysis, A.L.Cauchy (1789-1857), indeed felt comfortabe in the complex domain but the proof we have here utilizes very little the powerful features that come along in transition from real to complex numbers. The proofs by Liouville (1809-1882) and R.P.Boas, Jr. (1912-1992) make a convincing argument that the complex plane and the theory of analytic functions form the natural setting for the theorem.

Real functions may or may not have derivatives. Furthermore, existence of a derivative in one point does not assure its existence anywhere else. A real function may have one or two or, for that matter, any finite number of derivatives. In the complex plane, existence of the limit Δf/ Δz, as Δz

approaches 0, leads to a host of features unheard of among real functions. Functions for which this limit (the derivative) exists in every point of an open domain are called analytic (in this domain.) Functions analytic in the whole plane are called entire. Polynomials are entire functions. Analytic functions have derivatives of any order which themselves turn out to be analytic functions. Both real u and imaginary v, components of analytic functions f(z) = u(z) + iv(z) are real valued harmonic functions which, like a travelling wave, are completely defined by their boundary values. For analytic functions, this property is expressed by the Cauchy Integral Formula.

Functions which are not entire have singularities in the finite plane. Entire functions have a singularity at infinity. The only ones that do not are constant. Polynomials have a pole at infinity. All entire functions with a pole of the same order behave in a similar manner. All polynomials of order n behave similarly to zn which has been exploited in somewhat different ways in the proofs by Cauchy and those taken from books by Birkhoff and MacLane and Courant and Robbins. One facet of this property found an expression for more general analytic functions in the form of a theorem proven by the French mathematician Eugene Rouche (1832-1910) in 1883: under certain conditions, the inequality |f(z) + g(z)| < |f(z)| valid for z on a simple closed curve, implies that inside the curve, analytic function f and g have the same number of zeros. Applied to polynomials P(z) of degree n and zn with the curve being a circle of sufficiently large radius, Rouche's theorem yields the Fundamental Theorem of Algebra.

(For a more complete treatment of the history of the FTA see the MacTutor History of Mathematics Archive.)

Perfect numbers are complex, complex numbers might be perfect Fundamental Theorem of Algebra: Statement and Significance What's in a proof? More about proofs Axiomatics Intuition and Rigor How to Prove Bolzano's Theorem Early attempts Proofs of the Fundamental Theorem of Algebra

o Remarks on Proving The Fundamental Theorem of Algebra

Sketch of a Proof by Birkhoff and MacLane Sketch of Second Proof (after Cauchy) Details of the Cauchy's Proof Note on the Extreme Value Theorem Sketch of Proof by the methods of the theory of Complex

Variables (after Liouville) Maximum Modulus Theorem

o A Proof of the Fundamental Theorem of Algebra: Standing on the shoulders of giants

o Yet Another Proof of the Fundamental Theorem of Algebra

o Fundamental Theorem of Algebra - Yet Another Proof o A topological proof, going in circles and counting

o A Simple Complex Analysis Proof o An Advanced Calculus Proof

Complex numbers are in a sense perfect while there is little doubt that perfect numbers are complex.

Starting from the tail, perfect numbers have been studied by the Ancients (Elements, IX.36). Euler (1707-1783) established the form of even perfect numbers. [Conway and Guy, p137] say this:

 Are there any other perfect numbers? ... All we know about the odd ones is that they must have at least 300 decimal digits and many factors. There probably aren't any!

Every one would agree it's rather a complex matter to write down a number in excess of 300 digits. Allowing for a pun, if there are odd perfect numbers they may legitimately be called complex. What about complex numbers in the customary sense? There is at least one good reason to judge them perfect. The Fundamental Theorem of Algebra establishes this reason and is the topic of the discussion below.

In the beginning there was counting which gave rise to the natural numbers (or integers): 1,2,3, and so on. In the space of a few thousand years, the number system kept getting expanded to include fractions, irrational numbers, negative numbers and zero, and eventually complex numbers. Even a cursory glance at the terminology would suggest (except for

fractions) the reluctance with which the new numbers have been admitted into the family.

The oldest known record of mathematical development comes from the Rhind Papyrus dated at about 1700 B.C. The scroll appears to be a practical handbook of Egyptian mathematics with solutions to some 85 problems mostly involving fractions. Except for 2/3 all other fractions had 1 in the numerator. So that, for example, 2/61 was written as 1/40 + 1/244 + 1/488 + 1/610. No wonder the document was headed "Directions for knowing all dark things."

Irrational numbers have been discovered by the Pythagoreans (c 500 B.C.) They have been so shocked the fellow who divulged the secret to the broad world is reported to have been drowned. (The American Heritage Dictionary lists the following synonyms: mentally ill, psychotic, paranoid, deranged, invalid, defective, indefensible and more in the same vein.)

Negative numbers have not been fully accepted until the 18th century. Gerolamo Cardano (1501-1576) who scorned them as numeri ficti, devoted two separate chapters in his Ars Magna, one to equations in the form x3 + mx = p and another to equations x3 = mx + p. Rene Descartes (1596-1650), the father of Analytic Geometry, referred to negatives as recines

fausses, or false roots.

The number zero has been invented by Hindus sometime before 825 A.D. when it was described (along with their positional system in base 10) by the Persian mathematician al-Khowarizmi. The Hindu's term for zero was sunya,

or void. [Ore] mentions that translated into Arabic this became as-sifi, which is the common root of the words zero and cipher. I would assume that the latter rather came from the Hebrew sfira (counting) or sifra (digit). [Ore]'s version has the virtue of suggesting that even in small things there may be found fundamental secrets.

During 11th and 12th centuries a number of European scholars went to Spain to study Arab learning. I would speculate many of them met with or used books by the great Jewish poet, mathematician, and Torah commentator Abraham ben Meir ibn Ezra (1092-1167) who wrote a book on Arithmetic in which he explained the Arab system of numeration and a zero [Ore. p166]. The most influential on the spread of Arab numerals and the use of zero in Europe was Liber abaci (1202 A.D.) by Leonardo Fibonacci of Pisa (c 1175-1250).

Imaginary numbers have been discovered by Cardano when solving cubic equations. He ran into what would amount in the present day notations as a square root of a negative number. For somebody who looked askance at negative quantities, their square roots were bound to appear quite illusory. In the true mathematical spirit, he went on and used them nonetheless in formulas like (5 + √-5) + (5 - √-5) = 10 where the result happened to be real.

Leonhard Euler (1707-1783) made complex numbers commonplace and the first proof of the Fundamental Theorem of Algebra was given by Carl Friedrich Gauss (1777-1855) in his Ph.D. Thesis (1799). He considered the result so important he gave 4 different proofs of the theorem during his life time.

References

1. J. H. Conway and R.K.Guy, The Book of Numbers, Springer-Verlag, NY, 1996. 2. W. Dunham, The Mathematical Universe, John Wiley & Sons, NY, 1994. 3. H. Eves, Great Moments in Mathematics Before 1650, MAA, 1983 4. J. R. Newman, The Rhind Papyrus, in The World of Mathematics, v1, Simon

and Schuster, NY, 1956 5. O. Ore, Number Theory and Its History, Dover Publications, 1976 6. J. A. Paulos, Beyond Numeracy, Vintage Books, 1992

Fundamental Theorem of Algebra

Statement and Significance

We already discussed the history of the development of the concept of a number. Here I would like to undertake a more formal approach. Thus, in the beginning there was counting. But soon enough people got concerned with equation solving. (If I saw 13 winters and my tribe's law allows a maiden to marry after her 15th winter, how many winters should I wait before being allowed to marry the gorgeous hunter who lives on the other side of the mountain?) The Fundamental Theorem of Algebra is a theorem about equation solving. It states that every polynomial equation over the field of complex numbers of degree higher than 1 has a complex solution. Polynomial equations are in the form

 P(x) = anxn + an-1xn-1 + ... + a1x + a0 = 0,

where an is assumed non-zero (for why to mention it otherwise?), in which case n is called the degree of the polynomial P and of the equation above.

ai's are known coefficients while x is an unknown number. A number a is a solution to the equation P(x) = 0 if substituting a for x renders it identity: P(a) = 0. The coefficients are assumed to belong to a specific set of numbers where we also seek a solution. The polynomial form is very general but often studying P(x) = Q(x) is more convenient.

To see how this works let's start with the counting numbers (the set N of numbers 1,2,3,...), and the simplest equation x + a = b. For example, x + 5 = 12 has a solution x = 12 - 5 = 7. Also, x + 4 = 20 has a solution x = 20 - 4 = 16. Further, x + 20 = 4 has a solution x = 4 - 20 = ?. Oops, there is a problem. What is 4-20 among the counting numbers? No number exists such that when added to 20 gives 4. We say that the set of counting numbers is not algebraically closed. Introduction of negative numbers (note: I do not explain here how to do this) eases the problem:

 Any equation x + a = b where a,b∈N has a solution x∈Z, where Z is the set of integers numbers (plus, minus whole numbers and zero).

Actually, we get a little more than expected. For, once we accepted the negatives, we have a stronger result:

 Any equation x + a = b where a,b∈Z has a

solution x∈Z.

Even if the coefficients are allowed to be negative, the equation still has a solution in Z. Now let's consider other equations over Z: 5x - 10 = 0. x = 10/5 = 2. Also, 11x + 132 = 0. x = -132/11 = -12. Further, 5x - 11 = 0.

x = 11/5 = ?. You see? This happened again. Is there an integer such that when multiplied by 5 gives 11? No, Z is not algebraically closed either. However, introduction of rational numbers Q seems to solve the problem:

 Any equation ax + b = 0 where a,b∈Z has a

solution x∈Q

Moreover, we again get a stronger result, viz.:

 Any equation ax + b = 0 where a,b∈Q has a

solution x∈Q

Even when the coefficients are taken to be rational the equation still has a rational solution. However, Q is still not algebraically closed. For there are equations with rational coefficients (e.g., x 2   =   2 ) that have no rational solution. This leads to the set R of real numbers. R is a big field but not yet algebraically closed: the innocently looking equation x2 + 1 = 0 with real coefficients has no real solution. So eventually we introduce complex number field P. We expect that polynomial equations with real coefficients unsolvable among reals will have complex solutions. This is indeed so. But the Fundamental Theorem of Algebra states even more. It states that our perseverance paid off handsomely. Not only equations with real coefficients have complex solutions. Every polynomial equation with complex coefficients has at least one complex solution. In other words, the field of complex

numbers is algebraically closed!

Finally! Complex numbers are really perfect for solving equations.

What's in a proof?

Ever since Thales (640-550 BC) proved that the angles at the base of an isosceles triangle are equal and that a diameter divides a circle into two equal parts, the idea of proof, deduction of facts from (apparantly) simpler facts, has established itself as the characteristic aspect of mathematics. Euclid's Elements with its very deliberate selection of the simplest facts - postulates (axioms) - from which he derived more complicated facts - theorems - in geometry, arithmetic, and stereometry, served as a mathematical bible for more than 2,000 years.

The ultimate centrality of proof in mathematics found its expression in the formalistic view on mathematics which imposed itself as the New Math into the school education in the last half of this century. The current tendency is to counterbalance the deductive nature of mathematics with its rich historical and humanist background of its development. One of the chief arguments for this reversal was that the formalistic approach does not reflect on the way real mathematics is created by real mathematicicans (even those who identify with the formalistic school.)

Among working mathematicians the notion that a proof often serves to validate an idea conceived intuitively, is quite acceptable. A good proof reveals the content and the context of the theorem. A good proof, by discovering new and unexpected relations between mathematical objects, may lead to further discoveries and new theories. A proof is more important than a theorem. Mathematicians always try to find more revealing ways to look at a given statement. So a theorem would be proven time and again

over a course of many years. (For a simple example, check the Pythagorean Theorem.)

Gian-Carlo Rota writes:

It is an article of faith among mathematicians that after a theorem

is discovered, other simpler proofs of it will be given until definitive

one is found. A cursory inspection of the history of mathematics

seems to confirm the mathematician's faith. The first proof of a

great many theorems is needlessly complicated. "Nobody blames

mathematician if the first proof of a new theorem is clumsy," said

Paul Erdös. It takes a long time, from a few decades to centuries,

before the facts that are hidden in the first proof are understood,

as mathematician informally say. This gradual bringing out of the

significance of a discovery takes the appearance of a succession of

proofs, each one simpler than the preceding. New and simpler

versions of a theorem stop appearing when the facts are finally

understood.

Unfortunate mathematicians are baffled by the word

"understanding," which they mistakenly consider to have a

psychological rather than a logical meaning. They would rather fall

back on familiar logical grounds. They will claim that the search for

reasons, for an understanding of the facts of mathematics can be

explained by the notion of simplicity. Simplicity preferably in the

mode of triviality, is substituted for understanding. But is simplicity

characteristic of mathematical understanding. What really happens

to mathematical discoveries that are reworked over the years?

And a little later

Most mathematics discovered before 1800, with the possible

exception of some very few statements in number theory, can

nowadays be presented in undergraduate courses, and it is not too

far-fetched to label such mathematics as simple to the point of

triviality.

The Fundamental Theorem of Algebra is one of this statements that withstood many failed attempts and that generated a considerable interest (that resulted in numerous proofs) even after it has been proven by Gauss in 1799 in his doctoral dissertation. What's interesting is that, although Gauss is universally (and deservedly) credited with the first correct proof of the theorem, he himself in one place found it necessary to express his convinction that one aspect of the proof can be made quite rigorous. In a footnote he wrote:

It seems to be sufficiently well demonstrated that an algebraic

curve can neither be suddenly interrupted, nor lose itself after an

infinite number of terms, and nobody, to my knowledge, has ever

doubted it. But, if anybody desires it, then on another occasion I

intend to give a demonstration which will leave no doubt.

In his life time Gauss returned three more times to the theorem, the last time in 1849. However, the fact he promised was eventually established by Bolzano (1781-1848) and Weierstrass (1815-1897). In his 1817 paper on the Intermediate Value Thereom, Bolzano wrote thus:

There are two propositions in the theory of equations of which it

could still be said, until recently, that a completely correct proof

was unknown. One is the proposition: that between any two values

of the unknown quantity which give results of opposite sign there

must always lie at least one real root of the equation. The other is:

that every algebraic rational integral function of one variable

quantity can be divided into real factors of first or second degree.

After several unsuccessful attempts by d'Alembert, Euler, de

Foncenex, Lagrange, Laplace, Klügel, and others at proving the

latter proposition Gauss finally supplied, last year, two proofs

which leave very little to be desired. Indeed, this outstanding

scholar had already presented us with a proof of this proposition in

1799, but it had, as he admitted, the defect that it proved a purely

analytic truth on the basis of a geometrical consideration. But his

two most recent proofs are quite free of this defect; the

trigonometric functions which occur in them can, and must, be

understood in a purely analytical sense.

References

1. P. J. Davis and R. Hersh, The Mathematical Experience, Houghton Mifflin Company, Boston, 1981

2. The History of Mathematics , ed J. Fauvel and J. Gray, The Open University, 1987

3. G.-C. Rota, Indiscrete Thoughts, Birkhauser, 1997 4. D. J. Struik, A Source Book in Mathematics, 1200-1800, Princeton University

Press, Third Printing, 1990

More About Proofs

1. The evidence or argument that compels the mind to accept an assertion as true.

2. The validation of a proposition by application of specified rules, as of induction or deduction, to assumptions, axioms, and sequentially derived conclusions.

3. A statement or an argument used in such a validation.

American Heritage Dictionary

Every one knows that mathematics is about proving theorems. Mathematics is the only deductive science and both the study and development of

mathematics revolve around deducing one thing from another. Several misconceptions involving the notion of proof are quite common:

1. A proof is a deduction of facts from other, simpler facts. After all, Euclid built the body of Geometry on the foundation of 5 simple facts.

In reality, many math facts have been proven to be equivalent. Equivalency means that each of the two facts can be derived from the other. In what sense then one is simpler? Proof may be about seeing something in a different light or finding unexpected links.

2. This is how one develops mathematics - picking simple facts and

deriving from them something new.

This is actually never the case. Most often a mathematician senses that something is true, believes in it and a proof is a way of communicating the idea by giving it a sound foundation and, perhaps, explaining its origins. A proof transforms an idea into a fact. Quite often, attempts to prove an idea lead to its refutal.

3. Whatever is the right definition of the proof (most of us feel it in the

guts when something is doubtful while something else is certain, i.e.

proven), every one agrees with the definition. The definition is

immutable.

This is not true for several reasons. For example, there always were mathematicians who feel uncomfortable with the notion of infinity. Using the Axiom of Choice Tarski and Banach have shown that it's

possible to split a tennis ball into a few parts that could be combined again only to produce a ball of the size of our Earth. Many feel that the axiom is not a reliable tool. However, most mathematicians would not hesitate to use it.

The notion of proof underwent very dramatic changes in the 19th century when, for many reasons, mathematicians began questioning their intuition. Euclid's Elements define a straight line as a breadthless length. In 1945, in a Russian math olympiad, an 8th grade boy who did not even attempt to solve but one problem received a first prize for a remark he submitted with an unfinished proof of that problem:

 

I spent much time trying to prove that a straight line can't intersect three sides of a triangle in their interior points but failed for, to my consternation, I realized that I have no notion of what a straight line is.

In the 19th century mathematicians who questioned the 2300 year old intuition about straight lines discovered non-Euclidean geometries that, in the 20th century, were incorporated by Einstein into his General Theory of Relativity.

If intuition permeates any mathematical activity, how is it possible to build a theory that prides itself in its rigor on top of something rather elusive and individual? A historic example will shed some light on how this happens.

Axiomatics

For more than 2000 years the notion persisted that Euclid has developed Geometry on the basis of 5 simple facts known as Postulates. The five were indeed simple (it's not my intention at this point to talk of the famous fifth postulate and its history) but were not the only facts Euclid used to construct Geometry. He gave definitions to the terms of Geometry and introduced a variety of Common Notions (e.g., If equals be added to equals, the wholes

are equal.) that would count nowadays among axioms along with the Postulates.

Each of the 13 books of the Elements starts with a new set of Definitions. For example, Book III one starts with

Equal circles are those the diameters of which are equal, or the radii of which are equal.

This says that wherever two shapes are drawn according to the definition of a circle (a line of points equidistant from a fixed point) with equal radii, the shapes are equal. Compare this to the Fourth Postulate:

All right angles are equal to one another.

By definition (#10), an angle is right if it's obtained at the intersection of two straight lines where two adjacent angles are equal. Again it says that, if one follows the rule of construction (drawing intersecting lines with equal adjacent angles), the result will be one and the same right angle. The distinction between a Definition in Book III and a combination of a Definition

and a Postulate in Book I is obscure. The lesson we learn is that Euclid, although a source of the axiomatic method, frequently in his proofs relied on intuition in addition to his axioms.

D.Hilbert around the turn of the century offered a modern day axiomatization of the plane geometry based on 16 axioms. What's more interesting and edifying is that Hilbert does not define the terms used in his axioms. Euclid says

Definition 1. A point is that which has no part.Definition 2. A line id breadthless length.Postulate 1. Through any two points there is a straight line.

Hilbert just states

Axiom I-1. Through any two distinct points there is always a straight line.

There is no definition of what is a point or a straight line. By the end of the 19th century it became an acceptable fact that the terms of a theory are defined by its axioms. All that may be said of objects of a theory is conveyed by its axioms. Realization that any definition beyond that implied by the axioms depends on human intuition and has no place in a formal theory was quite formative in the development of modern mathematics starting with the 19th century.

Another example of an early axiomatization is supplied by Newton's fundamental Philosophiae naturalis principia mathematica. The Second Law of Motion reads

... the change of motion is proportional to the impressed force ...

Hence, if no impressed force is present, the motion remains unchanged (which is also given separately as the Law of Inertia.) One may compare this with the preceding definition (IV):

... the impressed force is the action on a body that changes its state of rest or of uniform rectilinear motion

The definition and the axiom state exactly the same thing; for, if the force is defined as that which changes the state of motion, in the absence of force the latter remains unchanged. Ernest Mach in his 1883's Mechanics reformulated Newton's theory by defining its terms only through their usage in axioms. Subsequently and independent of the theory, he also explained the terms' intuitive meaning.

Another example will illustrate how hard is it to entirely ban intuition from a mathematical proof.

References

1. R. von Mises, in J.Newman, The World of Mathematics, Simon and Schuster, 1956

Intuition and Rigor

One of the torch bearers of the formalization attempts in the 19th century was the Czech analyst Bernhard Bolzano (1781-1848.) In his critique of the attempts to prove the Fundamental Theorem of Algebra, he wrote

The most common kind of proof depends on a truth borrowed from

geometry, namely, that every continuous line of simple curvature

of which the ordinates are first positive and then negative (or

conversely) must necessarily intersect the x-axis somewhere at a

point that lies in between those ordinates. There is certainly no

question concerning the correctness, nor indeed the obviousness,

of this geometrical proposition. But it is clear that it is an

intolerable offense against correct method to derive truths of pure

(or general) mathematics (i.e., arithmetic, algebra, analysis) from

considerations which belong to a merely applied (or special) part,

namely, geometry ...

By which he of course meant that reliance on the geometrical intuition is an unacceptable tool in deriving analytic truths. He clearly accepts the statement as true but objects to the fact of its being used offhandedly, as a self-evident truth. In the article, Bolzano proceeds to justify the statement that is variably now known as Bolzano's or the Intermediate Value Theorem.

His proof depends on the definition of continuity by Cauchy from which he derives the Sign Preserving Property of Continuous Functions. Assuming that at the left end of an interval the function is negative, he observes that it stays negative on a certain bounded set but not for the points near the second end of the interval. He continues

Now the theorem holds that whenever a certain property M

belongs to all values of a variable quantity i which are smaller than

a given value and yet not for all values in general, then there is

always some greatest value u, for which it can be asserted that all

i<u possess property M.

I am not critical of Bolzano; for it's easy to see faults from the distance of 180 years. I only want to demonstrate how painful the labors were in delivering the present day mathematics. Bolzano ends his proof with the following remark:

It is now only a question of the proof of the theorem mentioned.

The theorem is proved by showing that those values of i of which it

can be asserted that all smaller values possess property M and

those of which this cannot be asserted can be brought as near one

another as desired. Whence it follows, for anyone who has a

correct concept of quantity, that the idea of a greatest value i of

which it can be said that all below it possess property M is the idea

of a real, i.e., actual, quantity.

I have a feeling that the actual quantity is somehow related to a point on a (geometric) line. In any event, Richard Dedekind (1831-1916) might not have had the "correct concept of quantity" for in 1872 he published the first rigorous introduction into the theory of real numbers. The claimed theorem has been indeed proven on the foundation of Dedekind's Cuts.

The property claimed by the theorem is known as completeness of the set of real numbers. Depending on how deep one wants to make a presentation of mathematics, the theorem may as well be declared an axiom - the fact that requires no proof in that particular setting. However, in the footsteps of Bolzano, it would be "an intolerable offense against correct method" to refer to somebody's concept of quantity in a proof claimed to be rigorous.

References

1. The History of Mathematics, ed J.Fauvel and J.Gray, The Open University, 1987

How to Prove Bolzano's Theorem

(without any epsilons or deltas!)

Scott E. Brodie, 11/2/97

Let f(x) be a continuous function on the closed interval [a,b], with

f(a) < 0 < f(b).

We are to show that there is a real number c, between a and b, such that f(c) = 0.

CONTINUITY:

We first need to understand what is meant by a continuous function. This is usually taken to mean that for real numbers near the number x, the function values lie near f(x). This is usually stated in terms of a "challenge" and "answer" dialogue:

We say "f is continuous at x" if, for each open interval J containing f(x), we can find an open interval I containing x so that for each point y in I, f(y) lies in the interval J.

We say "f(x) is continuous" if it is continuous in the sense given above at every point x of its domain.

It is traditional to think of the "challenge intervals" (the J's) as "small", in which case the "answer intervals" (the I's) will usually have to be small, too, but we will not explicitly rely on this.

As a simple application of this notion of continuity, we may note the

"Sign-Preserving property of Continuous Functions":

If f is continuous at a, and f(a) < 0, then there is an open interval I containing a such that f(x) < 0 for every x in I.

For a proof, simply take the open interval (2f(a),0) for the challenge interval "J" in the definition of continuity. Of course, a similar statement holds for f(b) > 0, with an analogous proof.

COMMENT on COMPLETENESS:

Note that Bolzano's theorem fails over the field of rational numbers:

the continuous function

f(x) = x2 - 2

takes the value -2 for x = 0 and takes the value +2 for x = 2, yet there is no rational number for which x 2   -   2   =   0

This example serves as a reminder that a proof of Bolzano's theorem must include some reference to the "completeness" of the set of real numbers -- the property which distinguishes the real numbers from the rational numbers.

The Axiom of Completeness can be stated in many equivalent forms. One common version which is suitable for our purposes is the

"Least Upper Bound" axiom:

Every non-empty set of real numbers which is bounded above has a least upper bound.

PROOF of BOLZANO's THEOREM:

Let S be the set of numbers x within the closed interval from a to b where

f(x) < 0.

Since S is not empty (it contains a) and S is bounded (it is a subset of [a,b]), the Least Upper Bound axiom asserts the existence of a least upper bound,

say c, for S. We show that this number c satisfies the requirements of Bolzano's theorem:

There are three possibilities for f(c): either f(c) < 0, f(c) > 0, or f(c) = 0. We show the first two choices lead to contradictions.

Suppose f(c) < 0, so that c is a member of S. Then the Sign Preserving Property of Continuous Functions asserts the existence of an open interval I containing c where f takes only negative values -- that is, I is a subset of S. But I (and thus S) contains points greater than c, so c cannot be an upper bound (let alone the least upper bound) for S. This contradiction forces us to reject the possibility that f(c) < 0.

Suppose instead that f(c) > 0. The argument is similar but not identical to the previous case. Since f is continuous, there is an open interval H containing c where f takes only positive values. But H contains points less than c, which must also be upper bounds for S, since no point of H can lie in S. This contradicts our choice of c as the least upper bound for S, so we must reject the possibility that f(c) > 0.

We conclude that f(c) = 0, QED.

COMMENTS on the PROOF of BOLZANO's THEOREM:

What is "really" going on here?

One way to look at it is to examine some of the properties of open intervals:

An interval is a set of points with the property that if x, y are distinct points of the set, then every point between x and y is also a point of the set.

An open interval is the set of points which lie strictly between two distinct points, called the "endpoints" of the interval.

(A "closed interval" is the union of an open interval and its endpoints.)

An open interval cannot be "degenerate" - a single point is not an open interval.

For each point, c, in an open interval, there are points of the open interval which are greater than c and (other) points of the interval less than c.

For each point, c, in an open interval, there is a closed interval about c, contained (as a subset) in the open interval.

An open interval does not contain its least upper bound nor does it contain its greatest lower bound.

Now consider the union of two open intervals, say U=(a,b), and V=(c,d), where a < d. Comparing b and c, there are only three possibilities:

If b < c, then the union of U and V omits the closed interval [b,c];

If b = c, then the union U and V omits the point b;

If b > c, then the union of U and V forms an open interval (a,d), but the original intervals share the interval (c,b).

In summary, it is impossible for the union of two disjoint open intervals to form an interval.

Much of this behavior of open intervals can be generalized to the notion of "open sets". A set is "open" if it contains an open interval containing each of its points.

For example, in our proof of Bolzano's theorem, we showed, in

essence, that the set S of points where f takes on negative values is an open set (it contains an interval about each of its points). Similarly, The set T of points where f takes on positive values is an open set.

As with an open interval, an open set cannot contain its least upper bound, nor its greatest lower bound.

We can also generalize our observation about the union of two disjoint open intervals:

Theorem: It is impossible for the union of two disjoint, non-empty open sets to form an interval.

The proof is nearly a paraphrase of our proof of Bolzano's theorem:

Suppose to the contrary, that the union of disjoint non-empty open sets U, containing a, and V, containing b, forms an interval, I, with a < b. Consider the set S of points x such that the entire closed interval [a,x] lies in U. Since U is open, it contains an open interval about a, so the set S is not empty. Since U and V are disjoint, the set S is bounded above by b. Then the Least Upper Bound axiom asserts the existence of a least upper bound, say, c, for S. Clearly, a is less than c, and c is at most equal to b.

If c lies in U, then there must be an open interval about c, in U, such that for some point x in this interval, [a,x] is in U (otherwise there would be a smaller upper bound for S than c). But then we may extend the interval [a,x] beyond c without leaving U, in contradiction to our choice of c as an upper bound for S. Thus, c does not lie in U.

On the other hand, if c lies in V, then there is an interval about c of points of V, which includes upper bounds for S which are smaller than c, contradicting our choice of c as the least upper bound for S. Thus, c does not lie in V. In particular, c cannot coincide with b.

We have shown that the union of U and V fails to include the point c, which lies between a and b. Thus the union of U and V cannot be an interval, QED.

With these notions at hand, the logic behind Bolzano's theorem is immediately evident:

If the function f is continuous, with f(a) < 0 and f(b) > 0, then the sets S, of points x where f(x) < 0, and T, of points where f(x) > 0 are open sets. But if no point of [a,b] satisfies f(x)=0, then S and T exhaust [a,b], in contradiction to the previous argument.

In the language of "general topology", a set which cannot be decomposed into two disjoint, non-empty open sets is called "connected", and we have shown that Bolzano's theorem is in essence equivalent to the theorem that a closed interval is a connected set.

Early Attempts at FTA

A glance into early attempts even to formulate the Fundamental Theorem of Algebra correctly, demonstrates how much development of mathematics has been impeded by the absence of a proper language to describe mathematical concepts.

The first formulation of the theorem was due to Albert Girard (1595-1632), a native of Lorraine who worked in Holland and was the editor of the works of Simon Stevin. His L'invention nouvelle en l'algèbre (Amsterdam, 1629) bases its fame on its formulation of the fundamental theorem of algebra, which also shows that he already took complex number seriously. It must be noted that Struik's is the only book that mentions Girard. The rest in my library that mention the Fundamental Theorem of Algebra, start with d'Alembert. The

theorem has been for a long time known as the d'Alembert's Theorem. Here how Girard put it:

Theorem II. All equations of algebra receive as many solutions as

the denomination of the highest term shows, except the

incomplete, and the first faction of the solutions is equal to the

number of the first mixed, their second faction is equal to the

number of the second mixed; their third to the third mixed, and so

on, so that the last faction is equal to the closure, and this

according to the signs that can be observed in the alternate order.

("Incomplete polynomials" are the ones in which some coefficients vanish. "Mixed" are, in my belief, the monomials akxk.) He later wrote,

As to the incomplete equations, they have not always so many

solutions, nevertheless we can well explain the solutions whose

existence is impossible, and show wherein lies the impossibility

because of the defectiveness and incompleteness of the equation.

René Descartes (1596-1650), the father of Analytic Geometry, who made an attempt to formalize the whole of science by reducing it to the simplest and most obvious maxim "Cogito, ergo sum", still had troubles accepting negative numbers. In the following piece from his Géométrie the "false roots" he struggles with were unfortunate enough to need a minus sign in front of them.

Know then that in every equation there are as many distinct roots,

that is, values of the unknown quantity, as is the number of

dimensions of the unknown quantity.

Suppose, for example, x equal to 2, or x - 2 equal to nothing, and

again, x equal to 3, or x - 3 equal to nothing. Multiplying together

the two equations we have xx - 5x + 6 equal to nothing, or xx

equal to 5x - 6. This is an equation in which x has the value 2 and

at the same time has the value 3...

It often happens, however, that some roots are false, or less than

nothing. Thus, if we suppose x to represent the defect of a quantity

5, we have x + 5 equal to nothing which, multiplied by x³ - 9x² +

26x - 24, gives x4 - 4x3 - 19x2 + 106x - 120 equal to nothing, as an

equation having four roots, namely three true roots, 2, 3, 4, and

one false root, 5.

The French mathematician Jean d'Alembert (1717-1783) tried but failed to prove the theorem in 1748. In the words of W.Dunham, d'Alembert recognized the importance of such a statement and made a stab at a proof. His stab, unfortunately, was wide of the mark. Perhaps to accord him the honor of trying, the result was long known as "d'Alembert's theorem", in spite of the fact that he came nowhere near proving it. This seems somewhat akin to renaming Moscow after Napoleon simply because he tried to reach it.

Euler made an attempt to prove the theorem in 1749 in a paper Recherches

sur les racines imaginaires des équations. Gauss who opened his 1799 dissertation with a critical review of previous attempts, observed that Euler's proof lacks "the clarity which is required in mathematics." Euler proved a series of theorems for various polynomial degrees. First for n = 4, then n = 8, with special corollaries filling in the gaps. Then he stated the theorem for n = 2k. The proof leads to an equation of degree N = 32870 which he calls "oddly even" (impairement pair) meaning that N/2 is odd.

Euler believed that the proof is solid ("je crois qu'on n'y trouvera rien à redire"), but to strengthen the argument he gives extra proofs for degrees 6, 4k + 2, 8k + 4,..., 2kp, p an odd number. If Euler mixed a matter of belief with a matter of having proven something rigorously - who may not?

References

1. W.Dunham, Journey through Genius, Penguin Books, 1991 2. D.J.Struik, A Source Book in Mathematics, 1200-1800, Princeton University

Press, Third Printing, 1990

Proofs of the Fundamental Theorem of Algebra

In his first proof of the Fundamental Theorem of Algebra, Gauss deliberately avoided using imaginaries. When formulated for a polynomial with real coefficients, the theorem states that every such polynomial can be represented as a product of first and second degree terms. Second degree factors correspond to pairs of conjugate complex roots. Over the field of complex numbers a more elegant formulation is possible: every polynomial

is a product of first degree terms. The latter formulation is not only more elegant, it's also more revealing. Staying in the realm of real numbers it's hard to explain why and wherefrom quadratic terms appear. Complex numbers provide an immediate explanation.

Complex numbers indeed proved to be a natural setting for the theorem. But the realization of course did not come immediately. The father of the modern complex analysis, A.L.Cauchy (1789-1857), indeed felt comfortabe in the complex domain but the proof we have here utilizes very little the powerful features that come along in transition from real to complex numbers. The proofs by Liouville (1809-1882) and R.P.Boas, Jr. (1912-1992) make a convincing argument that the complex plane and the theory of analytic functions form the natural setting for the theorem.

Real functions may or may not have derivatives. Furthermore, existence of a derivative in one point does not assure its existence anywhere else. A real function may have one or two or, for that matter, any finite number of derivatives. In the complex plane, existence of the limit Δf/ Δz, as Δz approaches 0, leads to a host of features unheard of among real functions. Functions for which this limit (the derivative) exists in every point of an open domain are called analytic (in this domain.) Functions analytic in the whole plane are called entire. Polynomials are entire functions. Analytic functions have derivatives of any order which themselves turn out to be analytic functions. Both real u and imaginary v, components of analytic functions f(z) = u(z) + iv(z) are real valued harmonic functions which, like a travelling wave, are completely defined by their boundary values. For analytic functions, this property is expressed by the Cauchy Integral Formula.

Functions which are not entire have singularities in the finite plane. Entire functions have a singularity at infinity. The only ones that do not are constant. Polynomials have a pole at infinity. All entire functions with a pole of the same order behave in a similar manner. All polynomials of order n behave similarly to zn which has been exploited in somewhat different ways in the proofs by Cauchy and those taken from books by Birkhoff and MacLane and Courant and Robbins. One facet of this property found an expression for more general analytic functions in the form of a theorem proven by the French mathematician Eugene Rouche (1832-1910) in 1883: under certain conditions, the inequality |f(z) + g(z)| < |f(z)| valid for z on a simple closed curve, implies that inside the curve, analytic function f and g have the same number of zeros. Applied to polynomials P(z) of degree n and zn with the curve being a circle of sufficiently large radius, Rouche's theorem yields the Fundamental Theorem of Algebra.

(For a more complete treatment of the history of the FTA see the MacTutor History of Mathematics Archive.)

Sketch of a Proof by Birkhoff and MacLane

Consider the polynomial P(z) as a mapping from one copy of the complex plane (say, the z-plane) to another copy of the complex plane (say, the w-plane). Such a mapping transforms a circle |z| = r of radius r into a closed curve on the w-plane (see Figure 1). For very large values of r, the anz n term dominates, and the image is a closed curve which loops n times around the origin of the w-plane (see Figure 2). On the other hand, for very small values of r, we may neglect all the terms except amz m + a0, where am is the

coefficient with the lowest index m>0 which is not equal to zero, and the image loops m times around the point w = a0 (Figure 3). As r 0, the image collapses to the point a0. We now invoke the topological fact that if we start with r very large, and continuously reduce r to 0, simultaneously reducing the image in the w -plane from a large loop encircling the origin (likely more than once) to a point, we must necessarily encounter a stage where the image curve passes through the origin w = 0. Thus, we have shown that some point on the z-plane maps to w = 0 - that is, that there is a complex root of P(z). (For details, see Birkhoff & MacLaine, A Survey of Modern

Algebra, AK Peters, 1997)

Figure 1. Image of circle |z| = 2 under the

mapping P(z) = z3 - 2z2 + z - 1. Figure 2. Image of circle |z| = 20 under

the mapping P(z) = z3 - 2z2 + z - 1.

Figure 3. |z| = 0.1 under the mapping

P(z) = z3 - 2z2 + z - 1. Note the enlarged scale.

Figure 4. Image of circle |z| = 0.75 under

the mapping P(z) = z3 - 2z2 + z - 1. Apparently a root of this polynomial lies very

near this circle.

Sketch of Second Proof (after Cauchy)

Consider the quantity s(z) = |P(z)|, where

P(z) = anz n + an -1z n -1 + ... + a1z1 + a0

is a polynomial of degree n. Clearly, s(z)≠0. If s(z) assumes a global minimum value, say, σ, on C, then it suffices to prove that σ = 0, or equivalently, it suffices to derive a contradiction from the alternative assumption that σ > 0.

Suppose, then, that s(z0) = σ > 0. It is convenient to "re-center" the arithmetic by setting Q(z) = P(z + z0 ), so |Q(0)| = |P(z0 )| = σ, where σ is likewise a global minimum for |Q(z)|.

Expanding out the definition of |Q(z)|, we obtain a set of new coefficients: Q(z) = bnz n + bn -1z n -1 + ... + b1z1 + b0, where bn = an≠0, and b0 = P(z0 )≠0. Let m be the exponent of the lowest power of z in Q(z) whose coefficient bm is not zero. Now consider the behavior of Q(z) for points z whose absolute

value is very small, say the points z =  lying on a small circle centered at the origin of radius ρ. As observed above, as z sweeps once around this small circle, Q(z) closely approximates the behavior of bmzm + b0, which sweeps out a small circle (of radius |bm|ρm) around the point b0 = P(z0 ). (In fact, Q(z) sweeps around this circle m times.) By choosing ρ sufficiently small, we may ensure that the radius of this circle (that is, the magnitude of Q(z) - b0) is smaller than σ. Such a circle will necessarily intersect the line segment connecting the origin to the point b0 = P(z0 ), at a point, say Q(z1 ), nearer the origin than b0 = P(z0 ). But then |Q(z1 )| = |P(z1 + z0 )| <σ, contradicting our choice of σ as a global minimum of s(z) = |P(z)|. (Actually, Q(z) may only lie near this circle, not on it. Fortunately, the discrepancy consists of the remaining terms of Q(z) (if any), all of which contain powers of ρ higher than m. By taking ρ even smaller, if necessary, we can guarantee that this discrepancy does not wreck the geometry - see Figure 5.)

Figure 5.Behavior of Q(z) for "small" values of z. If z lies on the small circle of radius ρ centered at the origin of the z-plane, then Q(z)≈bmzm + b0, which is the larger circle in the figure, centered at Q(0) = b0, with radius |bm|ρm. For ρ sufficiently small, the remaining terms of (if any) of Q(z) sum to a vector whose magnitude (proportional to ρn, n >m, is less than the radius of the smaller circle in the figure.) As z sweeps out the small circle of radius ρ on the z-plane, the locus of Q(z) loops around Q(0) = b0 (in fact, m times), and necessarily intersects the vector drawn from the origin of the w-plane to Q(0).

Thus the assumption that σ  > 0 is untenable, and we conclude that σ = 0, that is, that P(z) has a complex root.

Details of the Proof by Cauchy

The task of wrestling this idea into a respectable proof has flustered a good many authors. For the sake of completeness, I paraphrase the best version I have yet found, from JMH Olmsted, Advanced Calculus Appleton Century Crofts, 1961, p 516:

Express the complex number -b0/bm in the form -b0/bm =  , and let

. Let z1 =  , where r is yet to be determined. Then for the dominant term under conditions of small r we have

.

We may thus rewrite Q(z1 ) in the form

.

Taking absolute values, and applying the triangle inequality, we have

We now choose r so small that the quantity [|b|m+1r + ... + |bn|rn-m] < |b0|/t, and also so small that rm<t (so that |t - rm| = t - rm). Then

This contradicts the assumption that  = |b0| is the global minimum value for Q(z), QED.

"But, that s(z) = |P(z)| assumes a global minimum value over C, we must prove thus." [I am here paraphrasing Euclid. The climactic conclusion of Euclid�s Elements is the construction of the five regular solids (tetrahedron, cube, octahedron, icosahedron and dodecahedron), and the demonstration that there can be no others (XIII. 18). There follows the anticlimactic proposition "But that the angle of the equilateral and

equiangular pentagon is a right angle and a fifth we must prove thus." Apparently, in the age of papyrus (before word-processing software), it was not so easy to repair an error of omission by inserting the material back at a more natural location.]

It is not true in general that the absolute value of a continuous (or even differentiable) function over C assumes a global minimum � in this regard, polynomials are special. For example, the exponential function ez = ex+iy = ex(cos(y) + isin(y)), which, we have already observed, has no roots over C, has magnitude |ex+iy| = ex, which is asymptotic to its greatest lower bound (0), but which never assumes 0 as a value.

In order to demonstrate that the absolute value of the polynomial P(z) assumes its minimal value, we examine its behavior for large |z|, as well as invoke the Extreme Value Theorem as applied to a real-valued function on a closed disk in C.

We have P(z) = anz n + an -1z n -1 + ... + a1z1 + a0. Factoring out zn, the

highest power of z, we have P(z) = zn(an + an -1z -1 + ... + a1z1-n + a0z-n). Taking absolute values, |P(z)| = |zn||an + an -1z -1 + ... + a1z1-n + a0z-n|. Now choose R sufficiently large that for each j<n,

(*)

whenever |z| > R.

Then for |z| > R, |an + an -1z -1 + ... + a1z1-n + a0z-n|≥|an|/2, and |P(z)| > Rnan /2 (See Figure 6.)

Figure 6. Behavior of |an + an-1z-1 + ... + a0z-n| for large |z|. an is shown as the purple

vector drawn from the origin. The remaining terms form the red zig-zag, whose vector sum is drawn in blue. By taking |z| suffciently large, this vector sum can be guaranteed to be smaller in magnitude than |an|/2, the radius of the circle with center at an, ensuring that |an + an-1z-1 + ... + a0z-n| for large |z| (the magnitude of the vector drawn

in green) is at least |an|/2.

Now choose any z2 such that |P(z2)| = u > 0. Choose R > 2(1 + u)/|an|, or larger, so that (*) is simultaneously satisfied . Then for |z| > R, we have |P(z)| > u.

Clearly, the minimal value, if any, of |P(z)| on the closed disc |z| ≤ R is the global minimum of |P(z)|, since |P(z2)| = u, but |P(z)| > u outside this disc. Now apply the Extreme Value Theorem to conclude that |P(z)| assumes a minimal value on the closed disc |z| ≤ R, which is a compact set. Then this minimal value is the global minimal value for |P(z)| over the entire complex plane, QED.

Note on the Extreme Value Theorem

The Extreme Value Theorem states that a continuous function from a compact set to the real numbers takes on minimal and maximal values on the compact set. (A compact subset of n-dimensional Euclidean space may be taken as any set that is closed (contains the limits of all convergent sequences made of points from the set) and bounded (contained within some finite n-dimensional "box").

We first prove the Bounded Value Theorem - the range of a continuous function on a compact set is bounded. Suppose not. Now proceed by successive bisection: bisect the original compact set (here is where we use the boundedness); on at least one piece, the function is unbounded. Bisect that piece again. (If the function is unbounded on both pieces, pick either one). Proceeding in this way, we obtain a nested sequence of boxes, of arbitrarily small maximum dimension, converging to a single point, say, c, in the original set (here is where we use the closedness). Since the function is continuous at c, there is a box, say B, containing c, such that for points within the box, all the function values differ by, say, no more than 1 from the value of the function at c - in other words, the function is bounded on the box B. But we claim to have produced a sequence of boxes around c of arbitrarily small size (some of which must necessarily fit entirely inside the box B) on

which the function is unbounded. This contradiction proves the Bounded Value Theorem

To prove the Extreme Value Theorem, suppose a continuous function f does not achieve a maximum value on a compact set. Since the function is bounded, there is a least upper bound, say M, for the range of the function. Consider the function g = 1/(f - M). Since f never attains the value M, g is continuous, and is therefore itself bounded. That implies that f does not get arbitrarily close to M, in contradiction to the choice of M as the least upper bound of the range of f. The same proof applies to the minimum value of f.

Non-elementary proofs of the Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra is deeply woven into the fabric of many aspects of mathematics, from which have emerged proofs with greatly differing viewpoints. Two such proofs, which are particularly well-known, are described briefly below, with references where details can be found.

Sketch of Proof by the methods of the theory of Complex Variables (after Liouville)

It is proved in the theory of Complex Variables that for a differentiable function f: C C,

where the integral is taken over a suitable closed curve enclosing a region where f and its derivatives are defined and continuous. By means of careful limiting arguments, it follows that the values of a differentiable function on a

suitable region of C are determined by its values on the boundary of the region, according to the Cauchy Integral Formula:

.

From this formula, a similar formula for the first derivative may be obtained:

From this integral formula, one obtains the Cauchy Estimate for the first derivative: if |f(z)| < M on the closed disc centered on z with radius R, then

From this estimate, one obtains Liouville�s Theorem: a bounded function differentiable on the entire complex plane is constant.

Now consider a non-constant polynomial P(z) of degree n, and suppose it has no roots. Then the reciprocal 1/P(z) is a continuous, differentiable function on the entire complex plane. Since, as we have shown above, if P(z) has no roots, |P(z)| takes on a global minimum value, 1/|P(z)| is bounded. By Liouville�s Theorem 1/P(z), and hence P(z), must be a constant, contradicting our choice of P(z). This proves the Fundamental Theorem of Algebra. (See, for example, Boas, RP, Invitation to Complex Analysis, Random House, New York, 1987, for details.)

In a recent email Professor Diego Vaggione of the National University of Cordoba, Argentina kindly drew my attention to a note of his that appeared in Colloquium Mathematicum not long ago. The note that presents a short proof of the Fundamental Theorem of Algebra follows (in an HTML rendition) the message from Professor Vaggione.

Dear professor Bogomolny:

I have visited your web site on mathematics. I found it very interesting. I am

enclosing a latex file of my paper "On the Fundamental Theorem of Algebra"

(Colloquium Mathematicum, Vol. 73, No. 2 (1997), 193-194) in which I show

that the clasical proof of the Fundamental Theorem of Algebra via Liouville

can be substantialy simplified. Perhaps you can include this proof at your

web site.

Best regards,

Diego Vaggione

COLLOQUIUM MATHEMATICUM

VOL. 73

1997NO.

2

 

ON THE FUNDAMENTAL THEOREM OF ALGEBRA

BY

DIEGO VAGGIONE (CÓRDOBA)

In most traditional textbooks on complex variables, the Fundamental Theorem of Algebra is obtained as a corollary of Liouville's theorem using elementary topological arguments.

The difficulty presented by such a scheme is that the proofs of Liouville's theorem involve complex integration which makes the reader believe that a proof of the Fundamental Theorem of Algebra is too involved. even when topological arguments are used.

In this note we show that such a difficulty can be avoided by giving a simple proof of the Maximum Modulus Theorem for rational functions and then obtaining the Fundamental Theorem of Algebra as a corollary. The proof obtained in this way is intuitive and mnemotechnic in contrast to the usual elementary proofs of the Fundamental Theorem of Algebra.

As usual we use C to denote the set of complex numbers. By D(a, ε) we

denote the set {z ∈ C: |z - a| < ε}.

LEMMA. Let f be a function such that f(D(a,ε)) is contained in a half plane

whose defining straight line contains 0. Let k ≥ 1. Then if the limit

limz→a f(z)/(z - a)k exists, it is 0.

Proof. Suppose limz → f(z)/(z - a)k = b≠0. Without loss of generality we can suppose that f(D(a, ε)) is contained in

the half plane {z: Re(z) ≥ 0} (replace f with cf for a suitable c ∈ C.) Let {zn: n≥ 1} be a sequence such that limn→∞zn = a and b(zn -

a)k is a negative real number, for every n ≥ 1. Thus we have

1 = limn→ε f(z)/b(zn - a)k

= Re limn→ε f(z)/b(zn - a)k

= limn→ε Re f(z)/b(zn - a)k ≤ 0,

which is absurd. Q.E.D.

MAXIMUM MODULUS THEOREM FOR RATIONAL FUNCTIONS. Let R(z) = p(z)/q(z), with p, q complex polynomials without common

factors. Suppose there exists a ∈ C such that q(a) ≠ 0 and |R(z)| ≤ |R(a)|, for every z ∈ D(a,ε), with ε > 0. Then R is a constant

function.

Proof. Suppose that R is not constant. Since p1(z) = q(a)p(z) - p(a)q(z) has a zero at z = a, there exists an integer k ≥ 1 and a

polynomial c(z) such that p1(z) = (z - a)kc(z) and c(a) ≠ 0. Thus

(R(z) - R(a))/(z - a)k = p1(z)/[q(a)q(z)(z - a)

k] = c(z)/[q(a)q(z)]

and therefore

limz→a (R(z) - R(a))/(z - a)k ≠ 0.

Since |R(z)|≤|R(a)| for every z ∈ D(a, ε), f(z) = R(z) - R(a) satisfies the hypothesis of the above lemma (make a picture). Thus we

arrive at a contradiction. Q.E.D.

FUNDAMENTAL THEOREM OF ALGEBRA. A polynomial with no zeros is constant.

Proof. Suppose that p(z) is not constant and p(z) ≠ 0 for every z ∈ C. Since limn → ∞ |p(z)| = ∞, there exists a ∈ C such that |p(a)|

≤ |p(z)| for all z ∈ C. Thus applying the Maximum Modulus Theorem to the rational function 1/p(z), we arrive at a contradiction.

Q.E.D.

Acknowledgement. I would like to thank María Elba Fasah for her assitance with the linguistic aspects of this note.

Facultad de Mathemática, Astronomía y Física (FAMAF)

Universidad Nacional de Córdoba

Ciudad Universitaria

Córdoba 5000, Argentina

Received 8 August 1996

Am Math Monthly 42(1935), 501-502

A Proof of the FundamentalTheorem of Algebra:

Standing on the shoulders of giants

R. P. Boas, Jr.

This note gives a proof, believed to be new, of the fundamental theorem of algebra; it is obtained by the use of the classical theorem of Picard: If there are two distinct values which a given entire function never assumes, the function is a constant. The proof is extremely simple and may be of interest as an application of Picard's theorem.

The fundamental theorem of algebra may be formulated as follows: An arbitrary polynomial,

f(z) = z n + a1z n -1 + ... + an -1z + an

where n is an integer > 0, and the ai are constants, has at least one zero (in the complex plane). We shall use in addition to Picard's theorem only the facts that f(z) is an entire function - hence, in particular, continuous - and that f(z) has a pole at infinity.

The proof is indirect. Suppose that f(z) is never zero. I say then that f(z) also fails to take on one of the values 1/k (k = 1,2,...). In fact, suppose that there are points zk such that f(zk ) = 1/k (k = 1,2...). Since f(z) has a pole at infinity, |f(z)| >1 uniformly outside some circle C. The points zk all lie within C, and hence have at least one limit point Z within C. Since f(z) is continuous,

This contradiction allows us to conclude that for some integer k, f(z) fails to take on the value 1/k. By Picard's theorem, f(z), never assuming the distinct values 0 and 1/k, must be constant, contrary to the hypothesis that the degree of f(z) was at least 1. This contradiction shows that f(z) must have at least one zero, and the proof is complete.

Yet Another Proof of the FundamentalTheorem of Algebra

R. P. Boas, Jr.

The following proof of the fundamental theorem of algebra by contour integration is similar to Ankeny's [1], but is simpler because it uses

integration around the unit circle (which is usually the first application of contour integration) instead of integration along the real axis; thus there is no need to discuss the asymptotic behavior of any integrals.

Let P(z) be a nonconstant polynomial; we are to show that P(z) = 0 for some

z. We may suppose P(z) real for real z. (Indeed, otherwise let be the polynomial whose coefficients are the conjugates of those of P(z) and

consider .) Suppose then that P(z) is real for real z and is never 0; we deduce a contradiction. Since P(z) does not either vanish or change sign for real z, we have

(1)

But this integral is equal to the contour integral

(2)

where Q(z) = znP(z + z-1) is a polynomial. For z ≠ 0, Q(z)≠0; in addition, if an is the leading coefficient in P(z), we have Q(0) = an ≠0. Since Q(z) is never zero, the integrand in (2) is analytic and hence the integral is zero by Cauchy's theorem, contradicting (1).

Reference

1. N.C. Ankeny, One more proof of the fundamental theorem of algebra, this Monthly, 54 (1947) 464.

Fundamental Theorem of AlgebraYet Another Proof

Anindya Sen

Theorem. The Fundamental Theorem of Algebra. Let

 P(z) = a0zn + a1zn-1 + ... + akzn - k + ... + an

be a polynomial of degree n ≥ 1 with complex numbers ai as coefficients.

Then P has a root, i.e., there is a φ C such that P(φ) = 0.

We prove the theorem by showing that Image(P) = C.

We assume the standard result that a complex polynomial P: C → C is a

proper map, i.e., P-1(A) is compact whenever A ⊂ C is compact. (P is continuous, and |P(x)| → ∞ as |x| → ∞. Hence, if A C is closed and bounded, so is P-1(A). Hence, P is proper.)

Let f: U → R2 be a differentiable map of an open set U ⊂ R2 to R2. A point x ∈ U is said to be a regular point of f if Df(x): R2 → R2 is invertible. Otherwise, x

is said to be a critical point of f. A point y R2 is said to be a critical value of f if it is the image of a critical point.

With this notation in mind, we first prove

Lemma 1. Let K be the set of critical values of P. Then K and P-1(K) are both

finite subsets of C.

Proof: The critical points of P are the points at which P�(z) = 0. Since P� is a polynomial of degree n � 1, there are at most n � 1 critical points. Since each critical value is the image of a critical point, K has at most n � 1 points. Now each critical value has at most n inverse images, hence, P�(K) has at most n(n � 1) points. (We use the fact that a complex polynomial of degree k has at most k roots. The proof of this result does not use the fundamental theorem of algebra.)

Lemma 2. Let X = C \ P-1(K) and Y = C \ K. Then P(X) = Y.

Proof. Lemma 1 ensures that both X and Y are open connected subsets of C. Also, observe that all points in X are regular points of P, i.e., DP(x) is invertible for all x X.

Since P: C → C is proper and C is locally compact, it follows that Image(P) is closed in C. P(X) = Y ∩ Image(P). Hence, P(X) is closed in Y.

Let y P(X). Then, y = P(x) for some x ∈ X. Since x is a regular point, the inverse function theorem tells us that there are open neighbourhoods U, V of

x, y respectively, such that P: U → V is bijective. Hence every point y ∈ P(X)

has an open neighbourhood also contained in P(X). Hence, P(X) is open in Y. Since Y is connected it follows that P(X) = Y.

Now, by definition, K ⊂ Image(P) and Lemma 2 tells us that C \ K Image(P). Hence, Image(P) = C The proof of the theorem is complete.

The crucial idea in this proof is that the plane remains connected after removing finitely many points. All the other results used hold for polynomials from R to R as well.

Our approach to the fundamental theorem of algebra is similar to arguments used in [1] to investigate proper, smooth maps with non-negative Jacobian between connected, orientable manifolds.

REFERENCE

1. A. Nijenhuis and R. W. Richardson, Jr., A theorem on maps with non-negative jacobians, Michigan Math. J. 9 (1962) 173�176.

A topological proof

going in circles and counting

Let us suppose that the polynomial

f(z) = an z n + an -1 z n -1 + ... + a1z1 + a0

has no root, so that for every complex number z

f(z) ≠ 0.

On this assumption, if we now allow z to describe any closed curve in the x, y

- plane, f(z) will describe a closed curve Γ which never passes through the origin.

 

 Proof of fundamental theorem of algebra.

We may, therefore, define the order of the origin O with respect to the function f(z) for any closed curve C as the net number of complete

revolutions made by an arrow joining O to a point on the curve Γ traced out

by the point representing f(z) as z traces out the curve C. As the curve C we shall take a circle with O as center and with radius t, and we define the function φ(t) to be the order of O with respect to the function f(z) for the circle about O with radius t. Clearly φ(0) = 0, since a circle with radius 0 is a single point, and the curve Γ reduces to the point f(0) ≠ 0. We shall show in the next paragraph that φ(t) = n for large values of t. But the order φ(t) depends continuously on t, since f(z) is a continuous function of z. Hence we shall have a contradiction, for the function φ(t) can assume only integral values and therefore cannot pass continuously from the value 0 to the value n.

It remains only to show that φ(t) = n for large values of t. We observe that on a circle of radius z = t so large that

t > 1 and t > |a0| + |a1| + ... + |an - 1|,

we have the inequality

 

|f(z) - z n|    = |an-1 z n-1 + an-2 z n-2 + ... + a0|    ≤ |an-1||z|n-1 + |an-2||z|n-2 + ... + |a0|

   = tn-1[|an-1| + ... +|a0| /tn-1]    ≤ tn-1[|an-1| + ... +|a0|] < tn = |z|n.

Since the expression on the left is the distance between the two points zn and f(z), while the last expression on the right is the distance of the point zn from the origin, we see that the straight line segment joining the two points f(z) and zn cannot pass through the origin so long as z is on the circle of radius t about the origin. This being so, we may continuously deform the curve traced out by f(z) into the curve traced out by zn without ever passing through the origin, simply by pushing each point f(z) along the segment joining it to zn. Since the order of the origin will vary continuously and can assume only integral values during this deformation, it must be the same for both curves. Since the order for zn is n, the order for f(z) must also be n. This completes the proof.

Reference

1. R. Courant and H. Robbins, What is Mathematics?, Oxford University Press, 1966.

Fundamental Theorem of Algebra via Cauchy's Theorem

Theorem (Fundamental Theorem of Algebra) Every polynomial of degree n 1 with complex coefficients has a zero in C.

Proof Let p(z)=zn+an−1zn−1+ +a1z+a0  be a polynomial of degree n 1  and assume that p(z)=0  for all z   C.

By Cauchy's integral theorem we have

z=rdzzp(z)=2 ip(0) =0

where the circle is traversed counterclockwise. On the other hand,

z=rdzzp(z) 2 r max z=r1 zp(z) =2 min z=r p(z) 0

as r   (since p(z) z n 1− z an−1 − − z na0 ),  which is a contradiction.

Reference

A. R. Schep, A Simple Complex Analysis and an Advanced Calculus Proof of the Fundamental Theorem of Algebra, Am Math Monthly 116, Jan 2009, 67-68.

Fundamental Theorem of Algebra via Contour Integration

Theorem (Fundamental Theorem of Algebra) Every polynomial of degree n 1  with complex coefficients has a zero in C.

Proof Let p(z)=zn+an−1zn−1+ +a1z+a0  be a polynomial of degree n 1  and assume that p(z)=0  for all z   C.

Define g:[0 ) [0 2 ]  C by g(r )=1 p(rei )    Then the function g is continuous on (0) (0 2 ) satisfying g=ir r g

Define now F: [0 )  C by F(r)= 02 g(r )d  . Then by Leibniz's rule for differentiation under the integral sign we have for all r 0 

irF (r)=ir 02 r gd = 02 gd =g(r 2 )−g(r 0)=0

Hence F (r)=0for all r 0 . This implies that F constant on [0 ) with F(r)=F(0)=2 p(0)

=0. On the other hand, p(z) 0  as r  , which is a contradiction.

Reference

A. R. Schep, A Simple Complex Analysis and an Advanced Calculus Proof of the Fundamental Theorem of Algebra, Am Math Monthly 116, Jan 2009, 67-68.