I Proving the Existence of Roots in Complex Polynomials

[swampwiz]

The easiest proof uses topology, which has a nice intuitive feel, allowing folks to use aspects of it without going into the nitty-gritty.

First presume that these solutions exist in ℂ, then presume a path that x takes in which it is a circle of very large radius in ℂ, and thus which encircles all the solutions. The corresponding path of the polynomial output will also be in ℂ, and in the form large, wavy circle-ish share - i.e., the ultimate term will be a large circle, with the lesser terms generating the waviness - and such that the waviness is too small for the path to miss winding around the origin.

Now decrease the radius of the path of x, which will cause the polynomial path to become a smaller, relatively more wavier circle, and at some point, the waves will be relatively strong enough to touch the origin. Since the origin represents the value 0, this must mean that there is some value in the circle of x that is a solution of the polynomial, and so the polynomial can be factored into a pair of factors in which one is a linear term with the negative of this just found solution and a residual factor that has degree -1 from the original.

Now simply repeat the exercise using the residual factor; each iteration will result in another solution being found, leaving as the residual a corresponding lower degree polynomial. At some point the residual polynomial will be of degree 1, and thus the solution for it is trivially the free term. The net effect of all this is that the number of solutions that are found will correspond to the degree of the original polynomial.

QED

 

[mathwonk]

Just a question, re post #27. I don't see why the quadratic formula is purely algebraic. I.e. how does one prove that a square root exists, even for every positive real number, without analysis? e.g. how does one prove sqrt(2) exists without taking a least upper bound?

 

[fresh_42]

how does one prove sqrt(2) exists without taking a least upper bound?

I know two methods: Dedekind cuts, or Cauchy sequences. Are there others?

$\sqrt{2}$ might be a bad example since it is the length of a diagonal, hence exists. But what about $\sqrt[7]{\pi^e}$?

 

[mathwonk]

good points. to be more precise, i wondered how does one prove it without analysis, i.e. limits. dedekind cuts and cauchy sequences seem like analysis, and existence of "diagonals" do not seem to be very algebraic either. so my curiosity was about how one proves existence of roots of quadratic polynomials purely within algebra, and I was looking for an algebraic proof of existence of a square root. I agree that, if you take enough geometry axioms for granted to establish a theory of area, that euclid does it in his Elements, book 1. I still do not see any algebraic proof in that. In algebra, I assume we have integers and rationals, but I don't see how to even define real numbers algebraically, but maybe one can consider algebraic field extensions of the rationals, and say one has square roots in a field of form Q[X]/(f(X)), where f is a quadratic polynomial. But if one starts from say axioms for reals, it takes a limiting type of proof to establish even square roots exist. Even if one takes the algebraic field extension approach, it seems not so elementary to define a big enough field extension F to contain arbitrary square roots of all elements of F, without some kind of zorn lemma. not analysis maybe but pretty modern use of infinities. I was originally assuming we started from a definition of reals in some form, usually axiomatically via existence of lub's, or even complexes, and then tries to prove existence of square roots just by algebra. even in geometry you seem to have to assume that certain constructions with compass and straightedge do actually work, i.e. that certain circles do meet certain lines, which is a kind of completeness axiom, as used in analysis. still just rambling...

 

[martinbn]

I think when it comes to solvability in radicals, the ability to take roots is a given. For example given that you can solve any equation of the form $x^2=a$, can you solve the general quadartic equation? And the answer is yes. Here I assume characteristic different than $2$, otherwise one has to change the eqaution appropriatly.

 

[lavinia]

manipulated to solve for the roots,\Just a question, re post #27. I don't see why the quadratic formula is purely algebraic. I.e. how does one prove that a square root exists, even for every positive real number, without analysis? e.g. how does one prove sqrt(2) exists without taking a least upper bound?

Right. Purely algebraic means that the equation can be manipulated to solve for the roots. But it does not guarantee that the roots exist.

While I don't know the history, it seems that the Greek mathematicians ,before Archimedes anyway, used compass and straight edge to construct numbers. It wouldn't surprise me if they thought that the rationals were all of the possible numbers before the square root of 2 was constructed as the diagonal of a square.

These constructions seem not to use analysis in the sense of limits and bounds.

Given any number it's square root can be constructed with straight edge and compass.

In particular, $√b^2-4ac$ can be constructed. So if one can write down a quadratic equation with known coefficients, one can construct the roots. In some sense this is analysis free although still not purely algebraic in the sense of manipulating equations.

Your post makes me wonder if there are generalized methods of construction that produce cube roots and higher order roots.

 

[fresh_42]

Your post makes me wonder if there is are generalized methods of construction that produce cube roots and higher order roots.

This depends on what you allow to be added to compass and ruler. IIRC there are certain curves like an Archimedean spiral or so that enables e.g. trisection. But if I think about how compass and ruler are translated into a Galois extension, then it might be tough to decide (and prove) what a general construction method is.

 

[mathwonk]

@lavinia: Thank you. I think I understand your meaning now of solutions obtained by manipulating the coefficients, to mean that the solutions can be expressed using algebraic operations and nth roots, starting from the coefficients and rational numbers, i.e. that the equation is what we now call "solvable". My only reservation is that extraction of roots seems to me not an algebraic operation. Of course the statement that a^n = b can be regarded as an algebraic statement, but so can the statement that 3a^3 + ab +c = 0.

So I apparently misunderstood you. But I also seem to have less confidence that I understand clearly the meaning of certain familiar symbols. E.g. I myself do not feel I know precisely what number equals the square root of 2, although I also like the geometric representation as (the pair consisting of) a diagonal (and a side). But even as I say this, my skeptical self asks again, "I like visualizing those two segments, but how do I know there is a 'number' relating the side and the diagonal?" Defining the number to be that pair, seems quite abstract and modern.

I.e. it seems to me that in Euclid, line segments (even given a unit segment) did not have "lengths" thought of as algebraically manipulable numbers. To me this would require an arithmetic of segments as Hartshorne does, but Euclid did not do. The origins of this theory is attributed by Hartshorne to Hilbert and Enriques, thousands of years after Euclid. The introduction to Hartshorne's chapter 4 discusses this interesting and provocative topic. There is of course room for differing opinions as he acknowledges.

In my notes I show that if one does even begin to do this, and defines the product of two pairs of line segments to be equal if the rectangles constructed from them are equi-decomposable, then one can skip the proportionality theory of Euclid from chapter 5, and invoke the work of earlier chapters instead. The fact that Euclid does not do this suggests again to me that he did not envision this theory of "numbers" defined by segments.

Maybe I seem to be splitting hairs, but I think it is fundamental that Euclid did not have a notion of numbers more general than rationals, and that it was the work of much later generations to construct them from his foundations. Indeed to me this is why Euclid is so important to study first, becuse it leads the way to a conception of real numbers expanded out of geometry, one that we later generations tend to take for granted, but most students today, lacking a thorough familiarity with Euclidean gometry, do not understand well.

 

[fresh_42]

"I like visualizing those two segments, but how do I know there is a 'number' relating the side and the diagonal?" Defining the number to be that pair, seems quite abstract and modern.

If we stay with the Greeks, then numbers were always considered as relations of different line segments. We can bring them to overlap with compass and ruler and then have $\sqrt{2}$ as the stretching factor of $1.$ This is geometry, so maybe again not really algebra. Algebra would mean to write $\mathbb{Q}\subseteq \mathbb{Q}(a)$ with $a^2-2=0$. Hence $\sqrt{2}$ exists in any algebraic sense per construction/definition.

 

[WWGD]

If we stay with the Greeks, then numbers were always considered as relations of different line segments. We can bring them to overlap with compass and ruler and then have $\sqrt{2}$ as the stretching factor of $1.$ This is geometry, so maybe again not really algebra. Algebra would mean to write $\mathbb{Q}\subseteq \mathbb{Q}(a)$ with $a^2-2=0$. Hence $\sqrt{2}$ exists in any algebraic sense per construction/definition.

Maybe a slight aside, but there is a nice, short proof of the irrationality of $\sqrt 2$: using Eisenstein criterion, $x^2-2=0$ has no Rational solution.

 

[DaveE]

First my apologies for sullying a Math forum with a practical but inaccurate spin-off from the engineering world. You can just ban me now, LOL. Still, this might be useful for guessing exact factors.

Jump to the middle of this PDF (pg 53, I think) to see a method of approximating polynomial roots. This is from "Fundamentals of Power Electronics" by R.W. Erickson, but we both learned it from R.D. Middlebrook at Caltech originally.

 

[lavinia]

@lavinia:

Maybe I seem to be splitting hairs, but I think it is fundamental that Euclid did not have a notion of numbers more general than rationals, and that it was the work of much later generations to construct them from his foundations. Indeed to me this is why Euclid is so important to study first, becuse it leads the way to a conception of real numbers expanded out of geometry, one that we later generations tend to take for granted, but most students today, lacking a thorough familiarity with Euclidean gometry, do not understand it well.

I am not sure what qualifies as a theory of numbers. I would like to read your notes. In the compass and straight edge constructions it seems that what is really going on is the definition of points on a Euclidean line. It makes me think that the Greeks were trying to show how the intuitive idea of a straight line actually corresponds to something that can be constructed. The points define segments - pairs of points - which then can be overlapped. The relations of these overlaps then characterizes the line. Perhaps they thought that rational overlaps gave the entire line at least until the square root of 2 was discovered.

I wonder how the square root of 2 changed their idea of how space is constructed. And what did people think in the intervening two thousand years before Gauss finally classified all of the constructible points?

What you are saying seems to be that there is more to defining roots than as mere symbols that allow one to factor polynomials. The Greeks seemed to have believed the same thing. That is why I was wondering if there are generalized geometric construction methods that give roots of polynomials of arbitrary degree.

 

[fresh_42]

I think we may concentrate on the real part of the algebraic closure of the rationals to eliminate the distracting aspects of imaginary numbers. I read your [@mathwonk] question as: "Is there an algebraic method to construct the/a real zero of a rational polynomial of odd degree?"

The answer is in my opinion a "no". I see the situation as:

• geometry: partially existent by compass and ruler plus Galois
• numeric: Newton's algorithm
• set theory: Dedekind cuts
• topology: Cauchy sequences
• analysis: continuity
• algebra: change of signs and $\mathbb{Q}[x]/(p(x))$

This boils down to the question of what an algebraic number in algebra means. And algebra is the only branch on my list that doesn't really care about existence. It exists because we have a field extension of a positive degree. The methods I saw in van der Waerden's algebra book only mimic algebraically what is done analytically otherwise.

Therefore, in a way, your objection is right. There is no algebraic aspect in the fundamental theorem of algebra. Algebra simply doesn't bother. The algebraic closure is defined to carry the zeros. All that can be asked concerning the name FTA is, why the algebraic closure of the rationals is a subfield of the complex numbers.

And that is what we really discuss here: The complex numbers require an analytical (or topological) definition. They cannot be described by purely algebraic means because they are topologically closed. It is unfair to demand an algebraic version for a subfield. Ergo: you [@mathwonk] are right. The fundamental theorem of algebra cannot be proven within algebra because it already uses topology to even phrase it.

 

[mathwonk]

You know it just dawned on me that it seems in a way odd that I think of using the intermediate value theorem to prove a sqrt of 2 exists, or some sort of limiting process. I.e. I am assuming given the real numbers, which to me is the natural outgrowth of the points on the Euclidean line, imposing a completeness condition, and then deducing existence of roots.

But of course the field of "algebraic" numbers contains roots of all equations, and yet is not at all topologically complete, indeed it is countable. So I go at the question sort of backwards, constructing a natural, but unnecessarily large field, the reals, perhaps also complexes, and then backing up to the algebraic numbers. I.e. I think of the reals as a natural, geometrically motivated, field of numbers, and then show they contain certain algebraic numbers such as (positive) square roots.

Of course this to me is a natural response to lavinia's stated goal of saying just what number is the solution of a certain equation, i.e. which real number is the solution. I.e. when you ask me to show you by calculation with the coefficients, just what number is a solution, I understand that to mean show which among the given, accepted, family of numbers, namely the real numbers, defined topologically, is a solution.

It is quite a different flavored question to ask whether one can construct somehow a field of "numbers" in which there is a solution. These algebraically constructed fields of numbers, like Q[X]/(f(X)), have, to my geometric mind, a sort of questionable existence, being constructed to satisfy our algebraic requirements in a somewhat tautological way.

So we have maybe three approaches:
i) define the reals by completeness axiom, or construct them as cuts or cauchy sequences or infinite decimals, and deduce existence of certain solutions;
ii) give axioms for geometry, letting points on a line be representatives for numbers, and give geometric constructions for certain roots. Here lines and circles suffice to solve all quadratic equations.
iii) make an abstract algebraic definition/construction of a field extension of the rationals in which certain solutions exist.

If we take anyone of these, say the third, it remains somewhat non obvious to show that the abstract field we construct can be embedded in one of the other fields. E.g. to find an embedding of Q[X]/(X^2-2) in the reals. This embedding would seem to require the same topological proof of existence of a real sqrt(2) as we started from using limits, least upper bounds,... Starting from #2, we have lavinia's question challenging us to find more and more algebraic solutions via more exotic "constructions."

Actually this is rather wonderful. I tend to think of math as having 3 flavors, algebra, analysis, and geometry. This question then has a treatment in all three, and we can also relate them! What a fun question, somewhat deceptively quite "elementary".