Évariste Galois and His Theory

[fresh_42]

Galois died in a duel at the age of twenty. Yet, he gave us what we now call Galois theory. It decides all three ancient classical problems, squaring the circle, doubling the cube, and partitioning angles into three equal parts, all with compass and ruler alone. Galois theory also tells us that there is no general formula to solve the integer equation
$$
x^5+a_1x^4+a_2x^3+a_3x^2+a_4x+a_5=0
$$Continue reading...

 

[jedishrfu]

Great article @fresh_42!

Thanks for taking the time to write it and share with everyone on PF and beyond.

A brief note on origami and cubic equations:

While the classical trisecting an angle and doubling a cube were deemed impossible with only straight edge and compass. They were solvable via origami. Sadly the ancient mathematicians didn't know the art.

https://plus.maths.org/content/power-origami

 

[swampwiz]

Sorry, I still don't get this Galois-ian explanation.

BTW, I have figured out a different way that sloppily proves the unsolvability, and have posted it in one of the forums here. It uses the fact that a free-term polynomial has solutions that are the [ n ( n - 1 ) ]-th of the discriminant, and so any formulae for the solutions must have the discriminant tucked away somewhere - i.e., because if there is a continuous function solution, a small perturbation of the coefficients for a free-term polynomial must be of the form that uses a small perturbation of the discriminant. Then add in the fact that a general polynomial can be made monic, and then be substituted with an linear offset term that zeroes out the ( n - 1 )-th coefficient, leaving ( n - 1 ) # of coefficients, and so any continuous function would need to be nested terms of the sum of some value and a root of the deeper-nested sum, with each nesting removing a coefficient degree-of-freedom. The discriminant comes into play because the net root of it must be [ n ( n - 1 ) ].

For n = 2, the discriminant must be within a 2-root (i.e., quadratic formula)

For n = 3, the discriminant must be within a 6-root made up of the previous 2-root and a new 3-root (which this article shows)

For n = 4, the discriminant must be within a 12-root made up of the previous 6-root and a new 2-root (which it is for the quartic formula)

for n = 5, the discriminant must be within a 20-root made up of the previous 12-root and a new ... er ... 5/3 root ... OOPS, not possible!

 

[fresh_42]

This doesn't make any sense. Whatever you mean by a 5/3 root, $a^{p/q}=\sqrt[q]{a^p}$ which is a legit expression within any allowed solution. Nevertheless, general polynomials from degree 5 or higher cannot be reduced to such expressions.

 

[swampwiz]

This doesn't make any sense. Whatever you mean by a 5/3 root, $a^{p/q}=\sqrt[q]{a^p}$ which is a legit expression within any allowed solution. Nevertheless, general polynomials from degree 5 or higher cannot be reduced to such expressions.

I said it was sloppy. The root operations must be as per integers. The quadratic, cubic & quartic formulae all correspond to this.

Maybe I shall look over my notes and make this case better. I should say that I think that I completely understand the concept of Lagrangian resolvents.

 

[fresh_42]

Maybe I shall look over my notes and make this case better.

I think it would be more promising to study the theory than to save what already sounds lost.

 

[fresh_42]

I still don't get this Galois-ian explanation.

For any solution $x_0$ to $x^5+a_1x^4 +a_2x^3+a_3x^2+a_4x+ a_0=0$ there belongs an extension of the rational numbers $\mathbb{Q}+x_0\cdot \mathbb{Q}+x_0^2\cdot \mathbb{Q}+x_0^3\cdot \mathbb{Q}+x_0^4\cdot \mathbb{Q}.$ Every such extension has a group of functions, that do not affect $\mathbb{Q},$ but map $x_0$ to another solution $x_1.$

Whenever we can solve such an equation with $+\, , \,- \, , \, : \, , \, \cdot \, , \,\sqrt[5]{\, . \,}$ or other roots, then we get certain groups of those functions. And from five on, they simply do not exist anymore.Sorry, but it can get shorter and simpler than that.

 

[swampwiz]

I think it would be more promising to study the theory than to save what already sounds lost.

Uh, I thought the whole idea of the article was a quickie discussion of Galois Theory.

For someone that is not super-refined in mathematics, I think I can follow stuff as about as anyone else, especially on the topic of the impossibility of the quintic, which has been a bit of a bit of an obsession of mine once I had discovered that there were formulae for the cubic & quartic.

My understanding is that to properly understand the theory, one would need to master undergraduate abstract algebra first, then tackle Galois theory. The author of this article seemed to be trying to do a super quickie discussion on it, and I was just pointing out that it was a little TOO quick.

 

[swampwiz]

For any solution $x_0$ to $x^5+a_1x^4 +a_2x^3+a_3x^2+a_4x+ a_0=0$ there belongs an extension of the rational numbers $\mathbb{Q}+x_0\cdot \mathbb{Q}+x_0^2\cdot \mathbb{Q}+x_0^3\cdot \mathbb{Q}+x_0^4\cdot \mathbb{Q}.$ Every such extension has a group of functions, that do not affect $\mathbb{Q},$ but map $x_0$ to another solution $x_1.$

Whenever we can solve such an equation with $+\, , \,- \, , \, : \, , \, \cdot \, , \,\sqrt[5]{\, . \,}$ or other roots, then we get certain groups of those functions. And from five on, they simply do not exist anymore.Sorry, but it can get shorter and simpler than that.

Is this mapping you are talking about the same thing as Lagrange and his resolvent method? I understand how that method works, but that method doesn't prove that something is impossible.

 

[fresh_42]

Is this mapping you are talking about the same thing as Lagrange and his resolvent method? I understand how that method works, but that method doesn't prove that something is impossible.

I don't think so.

Those mappings are permutations of solutions, just like conjugation maps $i$ to $-i$ with complex numbers. Now permutation groups with $2, 3$ or $4$ elements are all commutative. In order for higher order solutions, we would also need commutative groups of those mappings because there is no specific order among the solutions. But form $5$ on, permutation groups are no longer commutative. This makes it impossible. The formal proof, of why we need commutative mappings, and the proof that there are no such groups, however, is where mathematics starts. That needs more preliminaries and preparations to show.

 

[mathwonk]

at least they have decomposition series with commutative quotient groups, i.e. are "solvable", up to permutations of 5 objects. no doubt what you meant.

 

[OscarCP]

I just have come across this interesting thread and reading it has raised in my mind a naïve question that probably has been asked about this mathematical theory already many times and already in many places, but not by me, until now:

According to the article on Origami geometry that can been accessed by the link in the Original Comment in this thread: "The power of origami" by Liz Newton.

[With my own clarifications for those who have not read it between square brackets]:

"Trisecting the angle

The seventh axiom [of Origami Geometry] is the key to both trisecting an angle and doubling the cube [both considered unsolvable, according to Galois' theory]. Let's start with the angle construction. By following the steps outlined below, it is possible to see how these simple axioms can enable the folder to perform an operation which eluded Euclid. (Diagrams courtesy Robert J. Lang.)
(NB: This method works for any angle less than 90°. There are other methods that work for larger angles.)"

Now here is my own difficulty in understanding why there is a problem with trisecting an angle with ruler and compass (although a compass is not needed for this -- and also with doubling the cube, but I won't get into that one here) that needed Galois to solve it.
What follows assumes this is a postulate of Origami Geometry, just as is stated by Liz Newton:

Axiom 7: Given one point p and two lines l1 and l2, there is a fold that places p onto l1 and is perpendicular to l2.

Comment and consequent question:

Please, notice that I am not wondering about the whole of Galois' theory concerning equations, only this application to geometry.

C: It seems quite possible to replace the foldings with straight lines made on paper, or state in a proof that this can be done with a ruler, using a ruler to draw the lines and making, or implying in writing, the foldings around them as 180 degree rotations, or else as mirror symmetries about the lines, as shown in a series of pictures that explain the various steps in Newton's article.

So, it would seem to me that one can do the above with a ruler and include the 180 degree rotations as shown in the diagrams in the article, and none of the above seems to me contradicts Euclid, including, as far as I understand these, none of the Origami postulates contradicts Euclid's own either. And no compass is necessary (could this be a problem?). Euclid might or might not have used 180 degree rotations or mirror symetries in his "Elements", but I can't remember anything in Euclid that rules them out. Those rotations imply a third dimension. And if remember correctly, Euclid had theorems about 3-D solids, angles between planes, etc.

Q: How is my comment wrong?

 

[fresh_42]

Q: How is my comment wrong?

This is not how it works.
a) We do not debunk theories from which we know they are wrong.
b) Your post is written in a way that does not allow analyzing where you are wrong.
c) It has already been said that angle trisection is possible with auxiliary means, e.g. a spiral.

I tell you what you must do:
1) Give a construction manual for $1,259921049894873164767210607278\ldots$
2) Prove with analytical means that you actually constructed $\sqrt[3]{2}.$
3) Find a journal that publishes your work.

If you manage to perform 1), then you will have used something else than a compass and a straightedge.
If you get to point 2), then you have made a mistake at point 1), or in your calculations at point 2).
If you find a journal for point 3), then it is no scientific journal and definitely does not have a peer review in place, since every single mathematician on Earth knows, that it is impossible.

 

[OscarCP]

This actually does not answer my simple naïve question.

I am not trying to debunk Galois' theory.

I was asking for an indication of what is wrong with my reasoning, not on a lecture on how to post a comment someone thinks I do not know how to do.

I do not see any reason for giving references to publications, as my comment is self-contained.

I imagine my comment is wrong, as Galois' theory is entirely correct. All I am asking is what is wrong with my comment, as I have made quite clear in my posting.

So I still hope someone who knows why is it wrong be kind enough to let me know just that.

 

[fresh_42]

This actually does not answer my simple question.

I know.

I was asking for an indication of what is wrong with my reasoning, not on a lecture on how to post a comment someone thinks I do not know about.

And I said that you have to put it into a form where people actually have a chance to answer your question. What you did, was write a wallpaper of text and request to find a stone in that labyrinth.

I do not see any reason for giving references to publications, as my comment is self-contained.

Yes. It is so full of uncertainties that it can be discussed endlessly. The less rigorous a text is, the longer it takes to work with it because your imprecise reasoning is full of emergency exits. There is a reason we do not debunk nonsense. It is a waste of time.

Doubling the cube mandates the construction of $\sqrt[3]{2}$.
You claim you can do it.
So I asked you to deliver.

Galois theory says it cannot be done.
So you actually demanded: I am not willing to study the theory, nor will I put my work in a correctable form, but apparently something in my thoughts contradicts this theory. Where?The burden of proof is on your site, not the other way around.

 

[OscarCP]

Mine is a naïve comment. You, who seem to assume some non-existent questionable intent in it and ask for irrelevant precison in a very simle question, that is not a criticism of something or other, made in a comment where I have written why I am asking it in terms that are clear enough to figure out what is wrong with my reasoning, something that is not obvious to me, but this fact does not mean I am right, which I fully assume I am not, then let me assume that you have no idea of of how to answer my question, so I am asking you now: please let tnis go, and maybe someone who can answer what is wrong with my simple question's reasoning will do so. I hope without making a big song and dance about nothing at all.

 

[fresh_42]

Mine is a naïve comment. You, who seem to assume some non-existent questionable intent in it and ask for irrelevant precison in a very simle question, that is not a criticism of something or other, made in a comment where I have written why I am asking it in terms that are clear enough to figure out what is wrong with my reasoning, something that is not obvious to me, but this fact does not mean I am right, which I fully assume I am not, then let me assume that you have no idea of of how to answer my question, so I am asking you now: please let tnis go, and maybe someone who can answer what is wrong with my simple question's reasoning will do so. I hope without making a big song and dance about nothing at all.

I do not see how you get from that seventh axiom to a construction order of the specific length $\sqrt[3]{2}.$ And what I don't see even less is, how that fold in axiom seven can be found by compass and ruler. The origami geometry simply provides existences that cannot be achieved otherwise.

 

[OscarCP]

I do not intend discussing this further unless it is with someone who will give me a direct answer to my question in its own terms. I have taken some time and care to formulate it as clearly as I can in such terms and deserve such an answer. What someone can't see is irrelevant to my question as formulated.

For example: are those origami's 180 degree turns, that I believe are the one explicit difference with how Euclidean geometry was formulated, but it seems to me still allowed by its axioms, the reason for my mistake, or is it something else of that sort?

In other words, I fully expect an answer to my question as I have posed it.

 

[OscarCP]

Oh, wait a moment: you might have answered my naïve question, after all.
I forgot that this is not about something being demonstrable using Euclidean geometry, which I suspect it is, but about doing something using only a ruler and, or a compass.

Although I would flip the page using the ruler.
But I am guessing that does not count, as it is too practical.
And probably it may be also hard to find a reference on such a flipping in the peer-reviewed literature.

 

[OscarCP]

Hmmm

 

[OscarCP]

Hmmm ... Or maybe that is not the answer, because: what about using not physical page turns, but reflections instead?

For example, using a transparent ruler and a compass, find an aleady defined point on the line of a new fold, based on the previously made construction, then draw with the ruler a line from the original point to this new line. This line shold be perpendicular by cnstruction to the new line. Now place the ruler flat on the paper with an edge along the new fold line and mark with a perfect spot of ink (this is an idealized construction, not a real one) where the point to be relfected is exactly under the ruler, then flip the ruler, so the ink on it marks on the paper its symmetrical, reflected point, and continue from there until the next fold comes along. Etc. At the end, if all this were right, the angle, whatever it is, has been divided in exactly three parts.

This reasoning also has to be wrong, but from what I just wrote in support of it, it would seem that any angle less than 90 degrees can be divided in three parts using only a ruler and a compass.

So what is wrong with it?

 

[fresh_42]

Although I would flip the page using the ruler.
But I am guessing that does not count, as it is too practical.
And probably it may be also hard to find a reference on such a flipping in the peer-reviewed literature.

The original problems do not know the third dimension. The entire construction has to be performed on a sheet of paper, or in the Greek sand if you like. Doubling the cube also simply means constructing its side length, not an actual cube.

The origami axioms establish the existence of certain quantities, called folds. The existences themselves are not the problem. Of course, $\sqrt[3]{2}$ does exist, and it can be approximated even with compass and ruler, but not constructed exactly.

Consider the angle $120°.$ If we could cut it into three equal parts, we would have constructed $40°.$ If we constructed $40°,$ then we could construct a regular nonagon since $40°$ would be its inner angle at the center of a circle. However, nonagons cannot be constructed by compass and straightedge. So even this simple angle of $120°$ cannot be cut into three without auxiliary tools.

A $n-$gon is constructible if and only if $\varphi(n)=2^k$ for some $k\in \mathbb{N}$ where $\varphi$ is the function that counts comprime numbers between $1$ and $n$.
$$
\varphi (n)=\left|\left\{a\in \mathbb{N}\,|\,1\leq a\leq n\wedge \operatorname{gcd}(a,n)=1\right\}\right|
$$

Here is an entire list of these auxiliary tools (origami included:
https://en.wikipedia.org/wiki/Angle_trisection#Other_methods

 

[OscarCP]

"The original problems do not know the third dimension. The entire construction has to be performed on a sheet of paper, or in the Greek sand. Doubling the cube also simply means constructing its side length, not an actual cube"

So that would also contradict the validity of my argument using reflections by flipinng the rule into the forbidden dimension. which is more than the definition of the prblem I was aware of in terms of being just about a rule and a compass.

So I would like to find out more about this.

Is there a not too technical discussion that says so much and one does not need to subscribe, revealing one's most intimate secrets in a form to be filled first, before being able to read it? If so, and I am pointed to it, thanks.

 

[fresh_42]

Is there a not too technical discussion that says so much and one does not need to subscribe, revealing one's most intimate secrets in a form to be filled first, before being able to read it? If so, thanks.

What do you mean? I find Wikipedia isn't too technical.

https://en.wikipedia.org/wiki/Angle_trisection
https://en.wikipedia.org/wiki/Nonagon
https://en.wikipedia.org/wiki/Constructible_polygon

My advice is to also use the German pages:

Wait, before you turn away. If you use Chrome as browser, then you can right-click on those German pages and translate them into English. The translation might not be perfect, but usually sufficient to read. And they are normally less technical. I liked the 2nd link.

https://de.wikipedia.org/wiki/Dreiteilung_des_Winkels
https://de.wikipedia.org/wiki/Konstruierbares_Polygon
https://de.wikipedia.org/wiki/Regelmäßiges_Polygon
https://de.wikipedia.org/wiki/Neuneck
https://de.wikipedia.org/wiki/Siebzehneck

 

[OscarCP]

You might hve been a bit too fast to reply. I wasn't quite done writing. Now I think my reply is finished. Have a look. It is about using only paper or sand.
Or not.

 

[fresh_42]

You might hve been a bit too fast to reply. I wasn't quite done writing. Now I think my reply is finished. Have a look. It is about using only paper or sand.
Or not.

Lol, same here. Reload the page please.

 

[OscarCP]

So far I have looked at the English version of Wikipedia. But it does not say there that no flipping, nor anything that assumes a third dimension is allowed, because everything is to be done on a flat surface with the ruler always flat on it, and that turning it with one edge upwards or downwards is expressly forbidden by the way the problem has been defined through the ages.

There is, maybe a hint of that in Wikipedia, in that all the solutions proposed that work, all those go beyond the use of only a simple straight ruler and a compass, and even so, all are also made entirely on a flat surface.

But flipping the ruler would have been quite feasible to people that had to trisect angles as part of their tasks using only a straight ruler and a compass, even way before the Trojan War.
And I have no reason to think that they could not have figured that out themselves, used the idea and got on with their job.

So defining the problem, back when, as limited entirely to working on a flat surface is unlikely to have been taken very seriously by anyone, I would think.

Later I'll have a look at those German Wikipedia ones.

Thanks.

 

[fresh_42]

Later I'll have a look at those German Wikipedia ones.

That translation trick is great. It multiplies the availability of Wikipedia pages. And they are not simple translations of one another, they are of different content. I also use them if I want to write a foreign name with strange letters or accents correctly.

Have a look at
https://www.physicsforums.com/threads/how-to-use-the-w-in-www.1062388/

 

[OscarCP]

Oh, I so do agree. I have used, just last week, "Google Translate" to translate a whole thirty-line poem from Spanish to English, for example, and I was quite surprised how much better this automatic translation has become compared to even two years ago. I changed a few things just because I liked them better my way, not because the translation was wrong.

And not just translating from Spanish, that here in the USA, is familiar to many and has considerable more use than, let's say Dutch. Because I also translate emails from a Dutch pen-pal the same way and the English text comes out quite clearly OK, judging also on how my replies, based on those translations and likewise translated from English to Dutch on the other side, have not caused, so far, any obvious misunderstandings.

It used to be hard to understand some translated sentences, or there were mistakes that were so obvious that could be corrected on the fly while reading the translation, but only if one was familiar already with the topic. Some of those mistakes were actually quite amusing.

But none, or maybe rarely, of that meta-translation by the reader seems to be necessary now. Which considerably increases how much more we can do now by using the Internet to learn and communicate even across former language barriers.
It's as if Douglas Adam's idea of the Babel fish has come true, only in a form much less yucky.