Tag: proof

An elementary proof of the insolvability of the quintic

When I was in middle school, I remember my teacher telling me, after I learned the quadratic formula, that there was a general formula for solving cubic and quartic equations, but no such formula existed for solving the quintic. This was also when I first heard the infamous story of young Galois’s death from a duel.

Using my profound middle-school logic, I took this story as a challenge to devise my own formula for solving the quintic. Naturally, my efforts came up short.

When I was in high school, with this obsession still fully intact, I attempted to read through the wonderful monograph Field Theory and Its Classical Problems. Here’s the MAA review of this book:

Hadlock’s book sports one of the best prefaces I’ve ever read in a mathematics book. The rest of the book is even better: in 1984 it won the first MAA Edwin Beckenbach Book Prize for excellence in mathematical exposition.

Hadlock says in the preface that he wrote the book for himself, as a personal path through Galois theory as motivated by the three classical Greek geometric construction problems (doubling the cube, trisecting angles, and squaring the circle — all with just ruler and compass) and the classical problem of solving equations by radicals. Unlike what happens in most books on the subject, all three Greek problems are solved in the first chapter, with just the definition of field as a subfield of the real numbers, but without even defining degree of field extensions, much less proving its multiplicativity (this is done in chapter 2). Doubling the cube is proved to be impossible by proving that the cube root of 2 cannot be an element of a tower of quadratic extensions: if the cube root of 2 is in a quadratic extension, then it is actually in the base field. Repeating the argument, we conclude that it is not constructible because it is not rational. A similar argument works for proving that trisecting a 60 degree angle is impossible. Of course, proving that duplicating the cube is impossible needs a different argument: chapter 1 ends with Niven’s proof of the transcendence of π.

After this successful bare-hands attack at three important problems, Chapter 2 discusses in detail the construction of regular polygons and explains Gauss’s characterization of constructible regular polygons, including the construction of the regular 17-gon. Chapter 3 describes Galois theory and the solution of equations by radicals, including Abel’s theorem on the impossibility of solutions by radicals for equations of degree 5 or higher. Chapter 4, the last one, considers a special case of the inverse Galois problem and proves that there are polynomials with rational coefficients whose Galois group is the symmetric group, a result that is established via Hilbert’s irreducibility theorem.

Many examples, references, exercises, and complete solutions (taking up a third of the book!) are included and make this enjoyable book both an inspiration for teachers and a useful source for independent study or supplementary reading by students.

As I recall, I made it successfully through the first couple of chapters but started to get lost with the Galois theory somewhere in the middle of Chapter 3. Despite not completing the book, this was one of the most rewarding challenges of my young mathematical life. Perhaps one of these days I’ll undertake this challenge again.

Anyway, this year I came across the wonderful article The Abel–Ruffini Theorem: Complex but Not Complicated in the March issue of the American Mathematical Monthly. The article presents a completely different way of approaching the insolvability of the quintic that avoids Galois theory altogether.

The proof is elementary; I’m confident that I could have understood this proof had I seen it when I was in high school. That said, the word “elementary” in mathematics can be a bit loaded — this means that it is based on simple ideas that are perhaps used in a profound and surprising way. Perhaps my favorite quote along these lines was this understated gem from the book Three Pearls of Number Theory after the conclusion of a very complicated proof in Chapter 1:

You see how complicated an entirely elementary construction can sometimes be. And yet this is not an extreme case; in the next chapter you will encounter just as elementary a construction which is considerably more complicated.

I believe that a paid subscription to the Monthly is required to view the above link, but the main ideas of the proof can be found in the video below as well as this short PDF file by Leo Goldmakher.

Adding by a Form of 0: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on adding by a form of 0 (analogous to multiplying by a form of 1).

Part 1: Introduction.

Part 2: The Product and Quotient Rules from calculus.

Part 3: A formal mathematical proof from discrete mathematics regarding equality of sets.

Part 4: Further thoughts on adding by a form of 0 in the above proof.

Sum of Three Cubes

I now have a new example of an existence proof to show my students.

Last year, mathematicians Andrew Booker and Andrew Sutherland found solutions to the following two equations: x^3 + y^3 + z^3 = 33 and x^3 + y^3 + z^3 = 42. The first was found by Booker alone; the latter was found by the collaboration of both mathematicians. These deceptively simple-looking equations were cracked with a lot of math and a lot of computational firepower. The solutions:

(8,866,128,975,287,528)³ + (–8,778,405,442,862,239)³ + (–2,736,111,468,807,040)³ = 33

$latex (–80,538,738,812,075,974)3 + 80,435,758,145,817,5153 + 12,602,123,297,335,6313 = 42$

At the time of this writing, that settles the existence of solutions of x^3 + y^3 + z^3 = n for all positive integers n less than 100. For now, the smallest value of n for which the existence of a solution is not known is n = 114.

For further reference, including links to the original articles by Booker and then Booker and Sutherland, please see:

Borwein integrals

When teaching proofs, I always stress to my students that it’s not enough to do a few examples and then extrapolate, because it’s possible that the pattern might break down with a sufficiently large example. Here’s an example of this theme that I recently learned:

No automatic alt text available.

Source: https://www.facebook.com/MathematicalMemesLogarithmicallyScaled/photos/a.1605246506167805.1073741827.1605219649503824/2080975208594930/?type=3&theater

For further reading:

My Favorite One-Liners: Part 108

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Today’s post marks the final entry in this series. When I first came up with the idea of listing some of favorite classroom quips, I thought that this series might last a couple dozen posts. To my surprise, it instead lasted for more than 100 posts. I guess that, in my 21-year teaching career, I’ve slowly developed my own unique lexicon for communicating mathematical ideas, and perhaps this parallels (on a decidedly smaller scale) what a radio talk show host (like local legend Randy Galloway, who was a sports reporter/commentator in the Dallas/Fort Worth area for many years before retiring) does to build rapport with his/her audience.

I’ll use this final one-liner near the end of the semester when it’s time for students complete their evaluations of my teaching. Back in days of yore, professors would take 10-15 minutes to pass out paper copies of these evaluations, students would complete them, and that would be the end of it. In modern times, however, paper evaluations have switched to electronic evaluations, which are perhaps better for the environment but tend to have a decidedly lower response rate than the old paper evaluations. Still, I value my students’ feedback. So I’ll tell them:

Please fill out the student evaluation; the size of my raise depends on this.

After the laughter settles down, I’ll tell them, “Who’s joking?” I can’t say this happens everywhere, but I can honestly say that my department’s executive committee does consider student evaluations of teaching when deciding on the quality of my teaching, and that partially determines the size of my annual merit raise. (The committee also considers other indicators of good teaching other than student evaluations.)

It’s important to note that I don’t tell my students to give me a good evaluation; I just ask them to fill it out and to be honest with their feedback. I also tell them, forgetting my raise, I also want to hear from them about how the semester went. If it went great, I want to know that. If it sucked, I also want to know that. However, if they think the class sucked, just writing “This class sucked” doesn’t give me a lot of information about how to fix things for the next time that I teach the course. So, if they have a criticism, I ask them to give me specific feedback so that I can consider their critiques.

A couple years ago, I served on my university’s committee for reconsidering the way that we conduct student evaluations of teaching. To my surprise, when I interviewed students in focus groups, there was a general consensus that students believed that their evaluations were a waste of time that didn’t actually contribute anything to the university — or if they did contribute something, they had no idea what it was. Ever since then, I’ve made a point of telling my students that their evaluations really do matter and can make a difference in future offerings of my courses.

My Favorite One-Liners: Part 101

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

I’ll use today’s one-liner when a choice has to be made between two different techniques of approximately equal difficulty. For example:

Calculate \displaystyle \iint_R e^{-x-2y}, where R is the region \{(x,y): 0 \le x \le y < \infty \}

There are two reasonable options for calculating this double integral.

  • Option #1: Integrate with respect to x first:

\int_0^\infty \int_0^y e^{-x-2y} dx dy

  • Option #2: Integrate with respect to y first:

\int_0^\infty \int_x^\infty e^{-x-2y} dy dx

Both techniques require about the same amount of effort before getting the final answer. So which technique should we choose? Well, as the instructor, I realize that it really doesn’t matter, so I’ll throw it open for a student vote by asking my class:

Anyone ever read the Choose Your Own Adventure books when you were kids?

After the class decides which technique to use, then we’ll set off on the adventure of computing the double integral.

This quip also works well when finding the volume of a solid of revolution. We teach our students two different techniques for finding such volumes: disks/washers and cylindrical shells. If it’s a toss-up as to which technique is best, I’ll let the class vote as to which technique to use before computing the volume.

My Favorite One-Liners: Part 99

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Today’s quip is a light-hearted one-liner that I’ll use to lighten the mood when in the middle of a complex calculation, like the following limit problem from calculus:

Let f(x) = 11-4x. Find \delta so that |f(x) - 3| < \epsilon whenever $|x-2| < \delta$.

The solution of this problem requires isolating x in the above inequality:

|(11-4x) - 3| < \epsilon

|8-4x| < \epsilon

-\epsilon < 8 - 4x < \epsilon

-8-\epsilon < -4x < -8 + \epsilon

At this point, the next step is dividing by -4. So, I’ll ask my class,

When we divide by -4, what happens to the crocodiles?

This usually gets the desired laugh out of the middle-school rule about how the insatiable “crocodiles” of an inequality always point to the larger quantity, leading to the next step:

2 + \displaystyle \frac{\epsilon}{4} > x > 2 - \displaystyle \frac{\epsilon}{4},

so that

\delta = \min \left( \left[ 2 + \displaystyle \frac{\epsilon}{4} \right] - 2, 2 - \left[2 - \displaystyle \frac{\epsilon}{4} \right] \right) = \displaystyle \frac{\epsilon}{4}.

Formally completing the proof requires starting with |x-2| < \displaystyle \frac{\epsilon}{4} and ending with |f(x) - 3| < \epsilon.