neverendingbooks http://www.neverendingbooks.org lieven le bruyn's blog Tue, 27 Jul 2010 09:52:36 +0000 en hourly 1 http://wordpress.org/?v=3.0 math & manic-depression, a Faustian bargain http://www.neverendingbooks.org/index.php/math-manic-depression-a-faustian-bargain.html http://www.neverendingbooks.org/index.php/math-manic-depression-a-faustian-bargain.html#comments Thu, 15 Jul 2010 15:06:33 +0000 lievenlb http://www.neverendingbooks.org/?p=3242

In the wake of a colleague’s suicide and the suicide of three students, Matilde Marcolli gave an interesting and courageous talk at Caltech in April : The dark heart of our brightness: bipolar disorder and scientific creativity. Although these slides give a pretty good picture of the talk, if you can please take the time to watch it (the talk starts 44 minutes into the video).

Courageous because as the talk progresses, she gives more and more examples from her own experiences, thereby breaking the taboo surrounding the topic of bipolar mood disorder among scientists. Interesting because she raises a couple of valid points, well worth repeating.

We didn’t can see it coming

We are always baffled when someone we know commits suicide, especially if that person is extremely successful in his/her work. ‘(S)he was so full of activity!’, ‘We did not see it coming!’ etc. etc.

Matilde argues that if a person suffers from bipolar mood disorder (from mild forms to full-blown manic-depression), a condition quite common among scientists and certainly mathematicians, we can see it coming, if we look for the proper signals!

We, active scientists, are pretty good at hiding a down-period. We have collected an arsenal of tricks not to send off signals when we feel depressed, simply because it’s not considered cool behavior. On the other hand, in our manic phases, we are quite transparent because we like to show off our activity and creativity!

Matilde tells us to watch out for people behaving orders-of-magnitude out of their normal-mode behavior. Say, someone who normally posts one or two papers a year on the arXiv, suddenly posting 5 papers in one month. Or, someone going rarely to a conference, now spending a summer flying from one conference to the next. Or, someone not blogging for months, suddenly flooding you with new posts…

As scientists we are good at spotting such order-of-magnitude-out-behavior. So we can detect friends and colleagues going through a manic-phase and hence should always take such a person serious (and try to offer help) when they send out signals of distress.

Mood disorder, a Faustian bargain

The Faust legend : “Despite his scholarly eminence, Faust is bored and disappointed. He decides to call on the Devil for further knowledge and magic powers with which to indulge all the pleasures of the world. In response, the Devil’s representative Mephistopheles appears. He makes a bargain with Faust: Mephistopheles will serve Faust with his magic powers for a term of years, but at the end of the term, the Devil will claim Faust’s soul and Faust will be eternally damned.”

Mathematicians suffering from mood disorder seldom see their condition as a menace, but rather as an advantage. They know they do their best and most creative work in short spells of intense activity during their manic phase and take the down-phase merely as a side effect. We fear that if we seek treatment, we may as well loose our creativity.

That is, like Faust, we indulge the pleasures of our magic powers during a manic-phase, knowing only too well that the devilish depression-phase may one day claim our life or mental sanity…

]]> http://www.neverendingbooks.org/index.php/math-manic-depression-a-faustian-bargain.html/feed 1 Bourbaki and the miracle of silence http://www.neverendingbooks.org/index.php/bourbaki-and-the-miracle-of-silence.html http://www.neverendingbooks.org/index.php/bourbaki-and-the-miracle-of-silence.html#comments Tue, 13 Jul 2010 19:49:04 +0000 lievenlb http://www.neverendingbooks.org/?p=3193

The last pre-war Bourbaki congress, held in september 1938 in Dieulefit, is surrounded by mystery. Compared to previous meetings, fewer documents are preserved in the Bourbaki archives and some sentences in the surviving notules have been made illegible. We will have to determine the exact location of the Dieulefit-meeting before we can understand why this had to be done. It’s Bourbaki’s own tiny contribution to ‘le miracle de silence’…

First, the few facts we know about this Bourbaki congress, mostly from Andre Weil‘s autobiography ‘The Apprenticeship of a Mathematician’.

The meeting was held in Dieulefit in the Drome-Provencale region, sometime in september 1938 prior to the Munich Agreement (more on this next time). We know that Elie Cartan did accept Bourbaki’s invitation to join them and there is this one famous photograph of the meeting. From left to right : Simone Weil (accompanying Andre), Charles Pison, Andre Weil (hidden), Jean Dieudonne (sitting), Claude Chabauty, Charles Ehresmann, and Jean Delsarte.

Failing further written documentation, ‘all’ we have to do in order to pinpoint the exact location of the meeting is to find a match between this photograph and some building in Dieulefit…

The crucial clue is provided by the couple of sentences, on the final page of the Bourbaki-archive document deldi_001 Engagements de Dieulefit, someone (Jean Delsarte?) has tried to make illegible (probably early on).

Blowing the picture up, it isn’t too hard to guess that the header should read ‘Décision du 22 septembre 1938′ and that the first sentence is ‘Le Bourbaki de 2e classe WEIL fera pour le 15 octobre’. The document is signed

Camp de Beauvallon, le 22.IX.38.
L’adjudant de jour
DIEUDONNE

Now we are getting somewhere. Beauvallon is the name of an hamlet of Dieulefit, situated approximately 2.5km to the east of the center.

Beauvallon is rather famous for its School, founded in 1929 by Marguerite Soubeyran and Catherine Krafft, which was the first ‘modern’ boarding school in France for both boys and girls having behavioral problems. From 1936 on the school’s director was Simone Monnier.

These three women were politically active and frequented several circles. Already in 1938 (at about the time of the Bourbaki congress) they knew the reality of the Nazi persecutions and planned to prepare their school to welcome, care for and protect refugees and Jewish children.

From 1936 on about 20 Spanish republican refugees found a home here and in the ‘pension’ next to the school. When the war started, about 1500 people were hidden from the German occupation in Dieulefit (having a total population of 3500) : Jewish children, intellectuals, artists, trade union leaders, etc. etc. many in the Ecole and the Pension.

Because of the towns solidarity with the refugees, none were betrayed to the Germans, Le miracle de silence à Dieulefit. It earned the three Ecole-women the title of “Juste” after the war. More on this period can be read here.

But what does this have to do with Bourbaki? Well, we claim that the venue of the 1938 Bourbaki congress was the Ecole de Beauvallon and they probably used Le Pension for their lodgings.

We have photographic evidence comparing the Bourbaki picture with a picture taken in 1943 at the Ecole (the woman in the middle is Marguerite Soubeyran). Compare the distance between door and window, the division of the windows and the ivy on the wall.

Below two photographs of the entire school building : on the left, the school with ‘Le Pension’ next to it around 1938 (the ivy clad wall with the Bourbaki-door is to the right) and on the right, the present Ecole de Beauvallon (this site also contains a lot of historical material). The ivy has gone, but the main features of the building are still intact, only the shape of the small roof above the Bourbaki-door has changed.

During their stay, it is likely the Bourbakis became aware of the plans the school had would war break out. Probably, Jean Delsarte removed all explicit mention to the Ecole de Beauvallon from the archives upon their return. Bourbaki’s own small contribution to Dieulefit’s miracle of silence.

]]> http://www.neverendingbooks.org/index.php/bourbaki-and-the-miracle-of-silence.html/feed 0 Who was ‘le P. Adique’? http://www.neverendingbooks.org/index.php/who-was-le-p-adique.html http://www.neverendingbooks.org/index.php/who-was-le-p-adique.html#comments Thu, 08 Jul 2010 11:45:35 +0000 lievenlb http://www.neverendingbooks.org/?p=3108

The Bourbaki Code

  1. The wedding invitation that nearly killed Andre Weil
  2. When was the Bourbaki wedding?
  3. Where is the Royal Poldavian Academy?
  4. Where was the Bourbaki wedding?
  5. Seriously now, where was the Bourbaki wedding?
  6. Who was ‘le P. Adique’?

Last year we managed to solve the first few riddles of the Bourbaki code, but several mysteries still remain. For example, who was the priest performing the Bourbaki-Petard wedding ceremony? The ‘faire part’ identifies him as ‘le P. Adique, de l’Ordre des Diophantiens’.

As with many of these Bourbaki-jokes, this riddle too has several layers. There is the first straightforward mathematical interpretation of the p-adic numbers \hat{\mathbb{Z}}_p being used in the study of Diophantine problems.

For example, the local-global, or Hasse principle, asserting that an integral quadratic form has a solution if and only if there are solutions over all p-adic numbers. Helmut Hasse was a German number theorist, held in high esteem by the Bourbaki group.

After graduating from the ENS in 1929, Claude Chevalley spent some time at the University of Marburg, studying under Helmut Hasse. Hasse had come to Marburg when Kurt Hensel (who invented the p-adic numbers in 1902) retired in 1930.

Hasse picked up a question from E. Artin’s dissertation about the zeta function of an algebraic curve over a finite field and achieved the first breakthrough establishing the conjectured property for zeta functions of elliptic curves (genus one).

Extending this result to higher genus was the principal problem Andre Weil was working on at the time of the wedding-card-joke. In 1940 he would be able to settle the general case. What we now know as the Hasse-Weil theorem implies that the number N(p) of rational points of an elliptic curve over the finite field Z/pZ, where p is a prime, can differ from the mean value p+1 by at most twice the square root of p.

So, Helmut Hasse is a passable candidate for the first-level, mathematical, decoding of ‘le P. adique’.

However, there is often a deeper and more subtle reading of a Bourbaki-joke, intended to be understood only by the select inner circle of ‘normaliens’ (graduates of the Ecole Normale Superieure). Usually, this second-level interpretation requires knowledge of events or locations within the 5-th arrondissement of Paris, the large neighborhood of the ENS.

For an outsider (both non-Parisian and non-normalien) decoding this hidden message is substantially harder and requires a good deal of luck.

As it happens, I’m going through a ‘Weil-phase’ and just started reading the three main Weil-biographies : Andre Weil the Apprenticeship of a Mathematician, Chez les Weil : André et Simone by Sylvie Weil and La vie de Simone Weil by Simone Petrement.

From page 35 of ‘Chez les Weil’ : “Après la guerre, pas tout de suite mais en 1948, toute la famille avait fini par revenir à Paris, rue Auguste-Comte, en face des jardins du Luxembourg.” Sylvie talks about the Parisian apartment of her grandparents (father and mother of Andre and Simone) and I wanted to know its exact location.

More details are given on page 103 of ‘La vie de Simone Weil’. The apartment consists of the 6th and 7th floor of a building on the Montagne Sainte-Geneviève. The Weils bought it before it was even built and when they moved in, in may 1929, it was still unfinished. Compensating this, the apartment offered a splendid view of the Sacre-Coeur, the Eiffel-tower, la Sorbonne, Invalides, l’Arc de Triomphe, Pantheon, the roofs of the Louvre, le tout Paris quoi…

As to its location : “Juste au-dessous de l’appartement se trouvent l’Ecole des mines et les serres du Luxembourg, avec la belle maison ancienne où mourut Leconte de Lisle.” This and a bit of googling allows one to deduce that the Weils lived at 3, rue Auguste-Comte (the W on the map below).

Crossing the boulevard Saint-Michel, one enters the 5-th arrondissement via the … rue de l’Abbe de l’Epee… We did deduce before that the priest might be an abbot (‘from the order of the Diophantines’) and l’Epee is just ‘le P.’ pronounced in French (cheating one egue).

Abbé Charles-Michel de l’Épée lived in the 18th century and has become known as the “Father of the Deaf” (compare this to Diophantus who is called “Father of Algebra”). Épée turned his attention toward charitable services for the poor, and he had a chance encounter with two young deaf sisters who communicated using a sign language. Épée decided to dedicate himself to the education and salvation of the deaf, and, in 1760, he founded a school which became in 1791 l’Institution Nationale des Sourds-Muets à Paris. It was later renamed the Institut St. Jacques (compare Rue St. Jacques) and then renamed again to its present name: Institut National de Jeunes Sourds de Paris located at 254, rue Saint-Jacques (the A in the map below) just one block away from the Schola Cantorum at 269, rue St. Jacques, where the Bourbaki-Petard wedding took place (the S in the map).

Completing the map with the location of the Ecole Normale (the E) I was baffled by the result. If the Weil apartment stands for West, the Ecole for East and the Schola for South, surely there must be an N (for N.Bourbaki?) representing North. Suggestions anyone?

Previous in series

]]> http://www.neverendingbooks.org/index.php/who-was-le-p-adique.html/feed 0 NeB not among 50 best math blogs http://www.neverendingbooks.org/index.php/neb-not-among-50-best-math-blogs.html http://www.neverendingbooks.org/index.php/neb-not-among-50-best-math-blogs.html#comments Tue, 01 Jun 2010 09:09:54 +0000 admin http://www.neverendingbooks.org/?p=2987

Via Tanya Khovanova I learned yesterday of the 50 best math blogs for math-majors list by OnlineDegree.net. Tanya’s blog got in 2nd (congrats!) and most of the blogs I sort of follow made it to the list : the n-category cafe (5), not even wrong (6), Gowers (12), Tao (13), good math bad math (14), rigorous trivialities (18), the secret blogging seminar (20), arcadian functor (28) (btw. Kea’s new blog is now at arcadian pseudofunctor), etc., etc. . Sincere congrats to you all!

NeverEndingBooks didn’t make it to the list, and I can live with that. For reasons only relevant to myself, posting has slowed down over the last year and the most recent post dates back from february!

More puzzling to me was the fact that F-un mathematics got in place 26! OnlineDegree had this to say about F-un Math : “Any students studying math must bookmark this blog, which provides readers with a broad selection of undergraduate and graduate concerns, quotes, research, webcasts, and much, much more.” Well, personally I wouldn’t bother to bookmark this site as prospects for upcoming posts are virtually inexistent…

As I am privy to both sites’ admin-pages, let me explain my confusion by comparing their monthly hits. Here’s the full F-un history

After a flurry of activity in the fall of 2008, both posting and attendance rates dropped, and presently the site gets roughly 50 hits-a-day. Compare this to the (partial) NeB history

The whopping 45000 visits in january 2008 were (i think) deserved at the time as there was then a new post almost every other day. On the other hand, the green bars to the right are a mystery to me. It appears one is rewarded for not posting at all…

The only explanation I can offer is that perhaps more and more people are recovering from the late 2008-depression and do again enjoy reading blog-posts. Google then helps blogs having a larger archive (500 NeB-posts compared to about 20 genuine Fun-posts) to attract a larger audience, even though the blog is dormant.

But this still doesn’t explain why FunMath made it to the top 50-list and NeB did not. Perhaps the fault is entirely mine and a consequence of a bad choice of blog-title. ‘NeverEndingBooks’ does not ring like a math-blog, does it?

Still, I’m not going to change the title into something more math-related. NeverEndingBooks will be around for some time (unless my hard-disk breaks down). On the other hand, I plan to start something entirely new and learn from the mistakes I made over the past 6 years. Regulars of this blog will have a pretty good idea of the intended launch date, not?

Until then, my online activity will be limited to tweets.

]]> http://www.neverendingbooks.org/index.php/neb-not-among-50-best-math-blogs.html/feed 4 Return to LaTeX http://www.neverendingbooks.org/index.php/return-to-latex.html http://www.neverendingbooks.org/index.php/return-to-latex.html#comments Fri, 12 Feb 2010 20:20:29 +0000 lievenlb http://www.neverendingbooks.org/?p=2965

To most mathematicians, a good LaTeX-frontend (such as TeXShop for Mac-users) is the crucial tool to get the work done. We use it to draft ideas, write papers and courses, or even to take notes during lectures.

However, after six years of blogging, my own LaTeX-routine became rusty. I rarely open a new tex-document, and when I do, I’d rather copy-paste the long preamble from an old file than to start from scratch with a minimal list of packages and definitions needed for the job at hand. The few times I put a paper on the arXiv, the resulting text resembles a blog-post more than a mathematical paper, here’s an example.

As I desperately need to get some math-writing done, I need to pull myself away from the lure of an ever-open WordPress admin browser-screen and reacquaint myself with the far more efficient LaTeX-environment.

Perhaps even my blogging will benefit from the change. Whereas I used to keep on adding to most of my tex-files in order to keep them up-to-date, I rarely edit a blog-post after hitting the ’publish’ button. If I really want to turn some of my better posts into a book, I need them in a format suitable for neverending polishing, without annoying the many RSS-feed aggregators out there.

Who better than Terry Tao to teach me a more proficient way of blogging? A few days ago, Terry announced he will soon have his 5th (!!) book out, after three years of blogging…

How does he manage to do this? Well, as far as I know, Terry blogs in LaTeX and then uses a python-script called LaTeX2WP ’a program that converts a LaTeX file into something that is ready to be cut and pasted into WordPress. This way, you can write, and preview, your post in LaTeX, then run LaTeX2WP, and post into WordPress whatever comes out.’ More importantly, one retains a pure-tex-file of the post on which one can keep on editing to get it into a (book)-publishable form, eventually.

Nice, but one can do even better, as Eric from Curious Reasoning worked out. He suggests to install two useful python-packages : WordPressLib “with this library you can control remotely a WordPress installation. Use of library is very simple, you can write a small scripts or full applications that allows you to automate publishing of articles on your blog/site powered by WordPress” and plasTeX “plasTeX is a LaTeX document processing framework written entirely in Python. It currently comes bundled with an XHTML renderer (including multiple themes), as well as a way to simply dump the document to a generic form of XML”. Installation is easy : download and extract the files somewhere, go there and issue a sudo python setup.py install to add the packages to your python.

Finally, get Eric’s own wplatex package and install it as explained there. WpLaTeX has all the features of LaTeX2WP and much more : one can add titles, tags and categories automatically and publish the post from the command-line without ever having to enter the taboo WordPress-admin page! Here’s what I’ve written by now in TeXShop

I’ve added the screenshot and the script will know where to find it online for the blog-version as well as on my hard-disk for the tex-version. Very handy is the iftex … fi versus ifblog … fi alternative which allows you to add pure HTML to get the desired effect, when needed. Remains only to go into Terminal and issue the command

wplpost -x http://www.neverendingbooks.org/xmlrpc.php ReturnToLatex.tex

(if your blog is on WordPress.com it even suffices to give its name, rather than this work-around for stand-alone wordpress blogs). The script asks for my username and password and will convert the tex-file and post it automatic.

]]> http://www.neverendingbooks.org/index.php/return-to-latex.html/feed 5 Where’s Bourbaki’s Escorial? http://www.neverendingbooks.org/index.php/wheres-bourbakis-escorial.html http://www.neverendingbooks.org/index.php/wheres-bourbakis-escorial.html#comments Mon, 08 Feb 2010 14:30:59 +0000 lievenlb http://www.neverendingbooks.org/?p=2565

Early 1936, Andre Weil and Evelyne Gillet made a pre-honeymooning trip to Spain and visited El Escorial. Weil was so taken by the place that he planned the next Bourbaki-conference to be held in a nearby college. However, the Bourbakis never made it to to Spain that summer as the Spanish Civil War broke out July 17th, a few weeks before the intended conference. Can we GEO-tag the exact location of Bourbaki’s “Escorial”?

As explained in the bumpy-road-post, Andre Weil and Evelyne Gillet became involved sometime in 1935. Early 1936, they made a pre-honeymooning trip to Spain and visited El Escorial. Weil was so taken by the place that he planned the next Bourbaki-conference to be held in a nearby college.

However, the Bourbakis never made it to to Spain that summer as the Spanish Civil War broke out July 17th, a few weeks before the intended conference. Still, the second Bourbaki-meeting remains often referred to as the ‘Escorial conference’. Can we GEO-tag the exact location of Bourbaki’s “Escorial”?

Claude Chevalley came up with a Plan-B and suggested they would use his parents’ place in Chançay as their venue. Chevalley’s father was a French diplomat and his house sure did possess a matching ‘grandeur’ as can be seen from the famous picture below, taken at the (second) Chançay meeting in 1937 (Weil to the left, Chevalley to the right and Weil’s sister Simonne standing).

Thanks to the Bourbaki archives we know that the meeting took place from september 16th to 28th, that each of them had to pay 16 francs for full pension and had to bring along their own sheets and towels.

But where exactly is this beautiful house? Jacques Borowczyk has written a nice paper Bourbaki et la touraine in which he describes the Bourbaki congresses of 1936 and 1937 at the Chevalley-house in Chançay and further those held in 1956, 1957 and 1959 in ‘hôtel de la Brèche’ in Amboise.

Borowczyk places the Chevalley house in the little hamlet of Chançay, called “La Massoterie”. The village files assert that in 1931 three people were living at La Massoterie : father Abel Chevalley, who took residence there after his retirement in 1931, his wife Marguerite and their son Claude. But, at the time of the Bourbaki congres in 1936, Marguerite remained the only permanent inhabitant. Sadly, Abel Chevalley, who together with Marguerite compiled the The concise Oxford French dictionary, died in 1934.

Usually when you know the name of the hamlet, of the village and add just to be certain ‘France’, Google Maps takes you there within metres. So, this was going to be a quick post, for a change… Well, much to my surprise, typing ‘La Massoterie, Chançay, France’ only produced the answer “We could not understand the location La Massoterie, Chançay, France”.

Did I spell it wrong? Or, did the name change over times? No, Googling for it the first hit gives you the map of a 10km walk around Chançay passing through la Massoterie!

Now what? Fortunately Borowczyk included in his paper an old map, from Napoleonic times, showing the exact location of La Massoterie (just above the flash-sign), facing the castle of Volmer. If you compare it with the picture below from present day Chançay (via Google earth) it is surprising how many of the landmarks have survived the changes over two centuries.

It is now easy to pinpoint the exact location and zoom into the Chavalley-house, and, you’re in for a small surprise : the place is called La Massotterie with 2 t’s…

Probably, Googles database is more reliable than the information provided by the village of Chançay, or the paper by Borowczyk as it is the same spelling as on the old Napoleonic map. Anyway, feel free to have a peek at Bourbaki’s Escorial yourself!

]]> http://www.neverendingbooks.org/index.php/wheres-bourbakis-escorial.html/feed 2 Lambda-rings for formula-phobics http://www.neverendingbooks.org/index.php/lambda-rings-for-formula-phobics.html http://www.neverendingbooks.org/index.php/lambda-rings-for-formula-phobics.html#comments Fri, 05 Feb 2010 19:25:42 +0000 lievenlb http://www.neverendingbooks.org/?p=2853

In 1956, Alexander Grothendieck (middle) introduced \lambda-rings in an algebraic-geometric context to be commutative rings A equipped with a bunch of operations \lambda^i (for all numbers i \in \mathbb{N}_+) satisfying a list of rather obscure identities. From the easier ones, such as

\lambda^0(x)=1, \lambda^1(x)=x, \lambda^n(x+y) = \sum_i \lambda^i(x) \lambda^{n-i}(y)

to those expressing \lambda^n(x.y) and \lambda^m(\lambda^n(x)) via specific universal polynomials. An attempt to capture the essence of \lambda-rings without formulas?

Lenstra’s elegant construction of the 1-power series rings ~(\Lambda(A),\boxplus,\boxtimes) requires only one identity to remember

~(1-at)^{-1} \boxtimes (1-bt)^{-1} = (1-abt)^{-1}.

Still, one can use it to show the existence of ringmorphisms \gamma_n~:~\Lambda(A) \rightarrow A, for all numbers n \in \mathbb{N}_+. Consider the formal ‘logarithmic derivative’

\gamma = \frac{t u(t)'}{u(t)} = \sum_{i=1}^\infty \gamma_i(u(t))t^i~:~\Lambda(A) \rightarrow \Lambda(A)

where u(t)' is the usual formal derivative of a power series. As this derivative satisfies the chain rule, we have

\gamma(u(t) \boxplus v(t)) = \frac{t (u(t)v(t))'}{u(t)v(t)} = \frac{t(u(t)'v(t)+u(t)v(t)'}{u(t)v(t))} = \frac{tu(t)'}{u(t)} + \frac{tv(t)'}{v(t)} = \gamma(u(t)) + \gamma(v(t))

and so all the maps \gamma_n~:~\Lambda(A) \rightarrow A are additive. To show that they are also multiplicative, it suffices by functoriality to verify this on the special 1-series ~(1-at)^{-1} for all a \in A. But,

\gamma((1-at)^{-1}) = \frac{t \frac{a}{(1-at)^2}}{(1-at)} = \frac{at}{(1-at)} = at + a^2t^2 + a^3t^3+\hdots

That is, \gamma_n((1-at)^{-1}) = a^n and Lenstra’s identity implies that \gamma_n is indeed multiplicative! A first attempt :

hassle-free definition 1 : a commutative ring A is a \lambda-ring if and only if there is a ringmorphism s_A~:~A \rightarrow \Lambda(A) splitting \gamma_1, that is, such that \gamma_1 \circ s_A = id_A.

In particular, a \lambda-ring comes equipped with a multiplicative set of ring-endomorphisms s_n = \gamma_n \circ s_A~:~A \rightarrow A satisfying s_m \circ s_m = s_{mn}. One can then define a \lambda-ringmorphism to be a ringmorphism commuting with these endo-morphisms.

The motivation being that \lambda-rings are known to form a subcategory of commutative rings for which the 1-power series functor is the right adjoint to the functor forgetting the \lambda-structure. In particular, if A is a \lambda-ring, we have a ringmorphism A \rightarrow \Lambda(A) corresponding to the identity morphism.

But then, what is the connection to the usual one involving all the operations \lambda^i? Well, one ought to recover those from s_A(a) = (1-\lambda^1(a)t+\lambda^2(a)t^2-\lambda^3(a)t^3+...)^{-1}.

For s_A to be a ringmorphism will require identities among the \lambda^i. I hope an expert will correct me on this one, but I’d guess we won’t yet obtain all identities required. By the very definition of an adjoint we must have that s_A is a morphism of \lambda-rings, and, this would require defining a \lambda-ring structure on \Lambda(A), that is a ringmorphism s_{AH}~:~\Lambda(A) \rightarrow \Lambda(\Lambda(A)), the so called Artin-Hasse exponential, to which I’d like to return later.

For now, we can define a multiplicative set of ring-endomorphisms f_n~:~\Lambda(A) \rightarrow \Lambda(A) from requiring that f_n((1-at)^{-1}) = (1-a^nt)^{-1} for all a \in A. Another try?

hassle-free definition 2 : A is a \lambda-ring if and only if there is splitting s_A to \gamma_1 satisfying the compatibility relations f_n \circ s_A = s_A \circ s_n.

But even then, checking that a map s_A~:~A \rightarrow \Lambda(A) is a ringmorphism is as hard as verifying the lists of identities among the \lambda^i. Fortunately, we get such a ringmorphism for free in the important case when A is of ‘characteristic zero’, that is, has no additive torsion. Then, a ringmorphism A \rightarrow \Lambda(A) exists whenever we have a multiplicative set of ring endomorphisms F_n~:~A \rightarrow A for all n \in \mathbb{N}_+ such that for every prime number p the morphism F_p is a lift of the Frobenius, that is, F_p(a) \in a^p + pA.

Perhaps this captures the essence of \lambda-rings best (without the risk of getting an headache) : in characteristic zero, they are the (commutative) rings having a multiplicative set of endomorphisms, generated by lifts of the Frobenius maps.

]]> http://www.neverendingbooks.org/index.php/lambda-rings-for-formula-phobics.html/feed 1 Seating the first few thousand Knights http://www.neverendingbooks.org/index.php/seating-the-first-few-thousand-knights.html http://www.neverendingbooks.org/index.php/seating-the-first-few-thousand-knights.html#comments Wed, 03 Feb 2010 16:31:46 +0000 lievenlb http://www.neverendingbooks.org/?p=2855

The Knight-seating problems asks for a consistent placing of n-th Knight at an odd root of unity, compatible with the two different realizations of the algebraic closure of the field with two elements. The first identifies the multiplicative group of its non-zero elements with the group of all odd complex roots of unity, under complex multiplication. The second uses Conway’s ‘simplicity rules’ to define an addition and multiplication on the set of all ordinal numbers.

The odd Knights of the round table-problem asks for a specific one-to-one correspondence between two realizations of ‘the’ algebraic closure \overline{\mathbb{F}_2} of the field of two elements.

The first identifies the multiplicative group of its non-zero elements with the group of all odd complex roots of unity, under complex multiplication. The addition on \overline{\mathbb{F}_2} is then recovered by inducing an involution on the odd roots, pairing the one corresponding to x to the one corresponding to x+1.

The second uses Conway’s ‘simplicity rules’ to define an addition and multiplication on the set of all ordinal numbers. Conway proves in ONAG that this becomes an algebraically closed field of characteristic two and that \overline{\mathbb{F}_2} is the subfield of all ordinals smaller than \omega^{\omega^{\omega}}. The finite ordinals (the natural numbers) form the quadratic closure of \mathbb{F}_2.

On the natural numbers the Conway-addition is binary addition without carrying and Conway-multiplication is defined by the properties that two different Fermat-powers N=2^{2^i} multiply as they do in the natural numbers, and, Fermat-powers square to its sesquimultiple, that is N^2=\frac{3}{2}N. Moreover, all natural numbers smaller than N=2^{2^{i}} form a finite field \mathbb{F}_{2^{2^i}}. Using distributivity, one can write down a multiplication table for all 2-powers.

The Knight-seating problems asks for a consistent placing of n-th Knight K_n at an odd root of unity, compatible with the two different realizations of \overline{\mathbb{F}_2}. Last time, we were able to place the first 15 Knights as below, and asked where you would seat K_{16}

K_4 was placed at e^{2\pi i/15} as 4 was the smallest number generating the ‘Fermat’-field \mathbb{F}_{2^{2^2}} (with multiplicative group of order 15) subject to the compatibility relation with the generator 2 of the smaller Fermat-field \mathbb{F}_2 (with group of order 15) that 4^5=2.

To include the next Fermat-field \mathbb{F}_{2^{2^3}} (with multiplicative group of order 255) consistently, we need to find the smallest number n generating the multiplicative group and satisfying the compatibility condition n^{17}=4. Let’s first concentrate on finding the smallest generator : as 2 is a generator for 1st Fermat-field \mathbb{F}_{2^{2^1}} and 4 a generator for the 2-nd Fermat-field \mathbb{F}_{2^{2^2}} a natural conjecture might be that 16 is a generator for the 3-rd Fermat-field \mathbb{F}_{2^{2^3}} and, more generally, that 2^{2^i} would be a generator for the next field \mathbb{F}_{2^{2^{i+1}}}.

However, an “exercise” in the 1978-paper by Hendrik Lenstra Nim multiplication asks : “Prove that 2^{2^i} is a primitive root in the field \mathbb{F}_{2^{2^{i+1}}} if and only if i=0 or 1.”

I’ve struggled with several of the ‘exercises’ in Lenstra’s paper to the extend I feared Alzheimer was setting in, only to find out, after taking pen and paper and spending a considerable amount of time calculating, that they are indeed merely exercises, when looked at properly… (Spoiler-warning : stop reading now if you want to go through this exercise yourself).

In the picture above I’ve added in red the number x(x+1)=x^2+1 to each of the involutions. Clearly, for each pair these numbers are all distinct and we see that for the indicated pairing they make up all numbers strictly less than 8.

By Conway’s simplicity rules (or by checking) the pair (16,17) gives the number 8. In other words, the equation x^2+x+8 is an irreducible polynomial over \mathbb{F}_{16} having as its roots in \mathbb{F}_{256} the numbers 16 and 17. But then, 16 and 17 are conjugated under the Galois-involution (the Frobenius y \mapsto y^{16}). That is, we have 16^{16}=17 and 17^{16}=16 and hence 16^{17}=8. Now, use the multiplication table in \mathbb{F}_{16} given in the previous post (or compute!) to see that 8 is of order 5 (and NOT a generator). As a consequence, the multiplicative order of 16 is 5×17=85 and so 16 cannot be a generator in \mathbb{F}_{256}. For general i one uses the fact that 2^{2^i} and 2^{2^i}+1 are the roots of the polynomial x^2+x+\prod_{j<i} 2^{2^j} over \mathbb{F}_{2^{2^i}} and argues as before.

Right, but then what is the minimal generator satisfying n^{17}=4? By computing we see that the pairings of all numbers in the range 16…31 give us all numbers in the range 8…15 and by the above argument this implies that the 17-th powers of all numbers smaller than 32 must be different from 4. But then, the smallest candidate is 32 and one verifies that indeed 32^{17}=4 (use the multiplication table given before).

Hence, we must place Knight K_{32} at root e^{2 \pi i/255} and place the other Knights prior to the 256-th at the corresponding power of 32. I forgot the argument I used to find-by-hand the requested place for Knight 16, but one can verify that 32^{171}=16 so we seat K_{16} at root e^{342 \pi i/255}.

But what about Knight K_{256}? Well, by this time I was quite good at squaring and binary representations of integers, but also rather tired, and decided to leave that task to the computer.

If we denote Nim-addition and multiplication by \oplus and \otimes, then Conway’s simplicity results in ONAG establish a field-isomorphism between ~(\mathbb{N},\oplus,\otimes) and the field \mathbb{F}_2(x_0,x_1,x_2,\hdots ) where the x_i satisfy the Artin-Schreier equations

x_i^2+x_i+\prod_{j &lt; i} x_j = 0

and the i-th Fermat-field \mathbb{F}_{2^{2^i}} corresponds to \mathbb{F}_2(x_0,x_1,\hdots,x_{i-1}). The correspondence between numbers and elements from these fields is given by taking x_i \mapsto 2^{2^i}. But then, wecan write every 2-power as a product of the x_i and use the binary representation of numbers to perform all Nim-calculations with numbers in these fields.

Therefore, a quick and dirty way (and by no means the most efficient) to do Nim-calculations in the next Fermat-field consisting of all numbers smaller than 65536, is to use sage and set up the field \mathbb{F}_2(x_0,x_1,x_2,x_3) by

R.< x,y,z,t > =GF(2)[]
S.< a,b,c,d >=R.quotient((x^2+x+1,y^2+y+x,z^2+z+x*y,t^2+t+x*y*z))

To find the smallest number generating the multiplicative group and satisfying the additional compatibility condition n^{257}=32 we have to find the smallest binary number i_1i_2 \hdots i_{16} (larger than 255) satisfying

(i1*a*b*c*t+i2*b*c*t+i3*a*c*t+i4*c*t+i5*a*b*t+i6*b*t+
i7*a*t+i8*t+i9*a*b*c+i10*b*c+i11*a*c+i12*c+i13*a*b+
i14*b+i15*a+i16)^257=a*c

It takes a 2.4GHz 2Gb-RAM MacBook not that long to decide that the requested generator is 1051 (killing another optimistic conjecture that these generators might be 2-powers). So, we seat Knight K_{1051} at root e^{2 \pi i/65535} and can then arrange seatings for all Knight queued up until we reach the 65536-th! In particular, the first Knight we couldn’t place before, that is Knight K_{256}, will be seated at root e^{65826 \pi i/65535}.

If you’re lucky enough to own a computer with more RAM, or have the patience to make the search more efficient and get the seating arrangement for the next Fermat-field, please drop a comment.

I’ll leave you with another Lenstra-exercise which shouldn’t be too difficult for you to solve now : “Prove that x^3=2^{2^i} has three solutions in \mathbb{N} for each i \geq 2.”

]]> http://www.neverendingbooks.org/index.php/seating-the-first-few-thousand-knights.html/feed 0 big Witt vectors for everyone (1/2) http://www.neverendingbooks.org/index.php/big-witt-vectors-for-everyone-12.html http://www.neverendingbooks.org/index.php/big-witt-vectors-for-everyone-12.html#comments Tue, 02 Feb 2010 13:00:30 +0000 lievenlb http://www.neverendingbooks.org/?p=2811

Next time you visit your math-library, please have a look whether these books are still on the shelves : Michiel Hazewinkel‘s Formal groups and applications, William Fulton’s and Serge Lange’s Riemann-Roch algebra and Donald Knutson’s lambda-rings and the representation theory of the symmetric group.

I wouldn’t be surprised if one or more of these books are borrowed out, probably all of them to the same person. I’m afraid I’m that person in Antwerp…

Lately, there’s been a renewed interest in \lambda-rings and the endo-functor W assigning to a commutative algebra its ring of big Witt vectors, following Borger’s new proposal for a geometry over the absolute point.

However, as Hendrik Lenstra writes in his 2002 course-notes on the subject Construction of the ring of Witt vectors : “The literature on the functor W is in a somewhat unsatisfactory state: nobody seems to have any interest in Witt vectors beyond applying them for a purpose, and they are often treated in appendices to papers devoting to something else; also, the construction usually depends on a set of implicit or unintelligible formulae. Apparently, anybody who wishes to understand Witt vectors needs to construct them personally. That is what is now happening to myself.”

Before doing a series on Borger’s paper, we’d better run through Lenstra’s elegant construction in a couple of posts. Let A be a commutative ring and consider the multiplicative group of all ‘one-power series’ over it \Lambda(A)=1+t A[[t]]. Our aim is to define a commutative ring structure on \Lambda(A) taking as its ADDITION the MULTIPLICATION of power series.

That is, if u(t),v(t) \in \Lambda(A), then we define our addition u(t) \boxplus v(t) = u(t) \times v(t). This may be slightly confusing as the ZERO-element in \Lambda(A),\boxplus will then turn be the constant power series 1…

We are now going to define a multiplication \boxtimes on \Lambda(A) which is distributively with respect to \boxplus and turns \Lambda(A) into a commutative ring with ONE-element the series ~(1-t)^{-1}=1+t+t^2+t^3+\hdots.

We will do this inductively, so consider \Lambda_n(A) the (classes of) one-power series truncated at term n, that is, the kernel of the natural augmentation map between the multiplicative group-units ~A[t]/(t^{n+1})^* \rightarrow A^*. Again, taking multiplication in A[t]/(t^{n+1}) as a new addition rule \boxplus, we see that ~(\Lambda_n(A),\boxplus) is an Abelian group, whence a \Z-module.

For all elements a \in A we have a scaling operator \phi_a (sending t \rightarrow at) which is an A-ring endomorphism of A[t]/(t^{n+1}), in particular multiplicative wrt. \times. But then, \phi_a is an additive endomorphism of ~(\Lambda_n(A),\boxplus), so is an element of the endomorphism-RING End_{\Z}(\Lambda_n(A)). Because composition (being the multiplication in this endomorphism ring) of scaling operators is clearly commutative (\phi_a \circ \phi_b = \phi_{ab}) we can define a commutative RING E being the subring of End_{\Z}(\Lambda_n(A)) generated by the operators \phi_a.

The action turns ~(\Lambda_n(A),\boxplus) into an E-module and we define an E-module morphism E \rightarrow \Lambda_n(A) by \phi_a \mapsto \phi_a((1-t)^{-1}) = (1-at)^{-a}.

All of this looks pretty harmless, but the upshot is that we have now equipped the image of this E-module morphism, say L_n(A) (which is the additive subgroup of ~(\Lambda_n(A),\boxplus) generated by the elements ~(1-at)^{-1}) with a commutative multiplication \boxtimes induced by the rule ~(1-at)^{-1} \boxtimes (1-bt)^{-1} = (1-abt)^{-1}.

Explicitly, L_n(A) is the set of one-truncated polynomials u(t) with coefficients in A such that one can find elements a_1,\hdots,a_k \in A such that u(t) \equiv (1-a_1t)^{-1} \times \hdots \times (1-a_k)^{-1}~mod~t^{n+1}. We multiply u(t) with another such truncated one-polynomial v(t) (taking elements b_1,b_2,\hdots,b_l \in A) via

u(t) \boxtimes v(t) = ((1-a_1t)^{-1} \boxplus \hdots \boxplus (1-a_k)^{-1}) \boxtimes ((1-b_1t)^{-1} \boxplus \hdots \boxplus (1-b_l)^{-1})

and using distributivity and the multiplication rule this gives the element \prod_{i,j} (1-a_ib_jt)^{-1}~mod~t^{n+1} \in L_n(A). Being a ring-qutient of E we have that ~(L_n(A),\boxplus,\boxtimes) is a commutative ring, and, from the construction it is clear that L_n behaves functorially.

For rings A such that L_n(A)=\Lambda_n(A) we are done, but in general L_n(A) may be strictly smaller. The idea is to use functoriality and do the relevant calculations in a larger ring A \subset B where we can multiply the two truncated one-polynomials and observe that the resulting truncated polynomial still has all its coefficients in A.

Here’s how we would do this over \Z : take two irreducible one-polynomials u(t) and v(t) of degrees r resp. s smaller or equal to n. Then over the complex numbers we have u(t)=(1-\alpha_1t) \hdots (1-\alpha_rt) and v(t)=(1-\beta_1) \hdots (1-\beta_st). Then, over the field K=\mathbb{Q}(\alpha_1,\hdots,\alpha_r,\beta_1,\hdots,\beta_s) we have that u(t),v(t) \in L_n(K) and hence we can compute their product u(t) \boxtimes v(t) as before to be \prod_{i,j}(1-\alpha_i\beta_jt)^{-1}~mod~t^{n+1}. But then, all coefficients of this truncated K-polynomial are invariant under all permutations of the roots \alpha_i and the roots \beta_j and so is invariant under all elements of the Galois group. But then, these coefficients are algebraic numbers in \mathbb{Q} whence integers. That is, u(t) \boxtimes v(t) \in \Lambda_n(\Z). It should already be clear from this that the rings \Lambda_n(\Z) contain a lot of arithmetic information!

For a general commutative ring A we will copy this argument by considering a free overring A^{(\infty)} (with 1 as one of the base elements) by formally adjoining roots. At level 1, consider M_0 to be the set of all non-constant one-polynomials over A and consider the ring

A^{(1)} = \bigotimes_{f \in M_0} A[X]/(f) = A[X_f, f \in M_0]/(f(X_f) , f \in M_0)

The idea being that every one-polynomial f \in M_0 now has one root, namely \alpha_f = \overline{X_f} in A^{(1)}. Further, A^{(1)} is a free A-module with basis elements all \alpha_f^i with 0 \leq i &lt; deg(f).

Good! We now have at least one root, but we can continue this process. At level 2, M_1 will be the set of all non-constant one-polynomials over A^{(1)} and we use them to construct the free overring A^{(2)} (which now has the property that every f \in M_0 has at least two roots in A^{(2)}). And, again, we repeat this process and obtain in succession the rings A^{(3)},A^{(4)},\hdots. Finally, we define A^{(\infty)} = \underset{\rightarrow}{lim}~A^{(i)} having the property that every one-polynomial over A splits entirely in linear factors over A^{(\infty)}.

But then, for all u(t),v(t) \in \Lambda_n(A) we can compute u(t) \boxtimes v(t) \in \Lambda_n(A^{(\infty)}). Remains to show that the resulting truncated one-polynomial has all its entries in A. The ring A^{(\infty)} \otimes_A A^{(\infty)} contains two copies of A^{(\infty)} namely A^{(\infty)} \otimes 1 and 1 \otimes A^{(\infty)} and the intersection of these two rings in exactly A (here we use the freeness property and the additional fact that 1 is one of the base elements). But then, by functoriality of L_n, the element u(t) \boxtimes v(t) \in L_n(A^{(\infty)} \otimes_A A^{(\infty)}) lies in the intersection \Lambda_n(A^{(\infty)} \otimes 1) \cap \Lambda_n(1 \otimes A^{(\infty)})=\Lambda_n(A). Done!

Hence, we have endo-functors \Lambda_n in the category of all commutative rings, for every number n. Reviewing the construction of L_n one observes that there are natural transformations L_{n+1} \rightarrow L_n and therefore also natural transformations \Lambda_{n+1} \rightarrow \Lambda_n. Taking the inverse limits \Lambda(A) = \underset{\leftarrow}{lim} \Lambda_n(A) we therefore have the ‘one-power series’ endo-functor \Lambda~:~\wis{comm} \rightarrow \wis{comm} which is ‘almost’ the functor W of big Witt vectors. Next time we’ll take you through the identification using ‘ghost variables’ and how the functor \Lambda can be used to define the category of \lambda-rings.

]]> http://www.neverendingbooks.org/index.php/big-witt-vectors-for-everyone-12.html/feed 2 The odd knights of the round table http://www.neverendingbooks.org/index.php/the-odd-knights-of-the-round-table.html http://www.neverendingbooks.org/index.php/the-odd-knights-of-the-round-table.html#comments Thu, 28 Jan 2010 20:48:59 +0000 lievenlb http://www.neverendingbooks.org/?p=2768

Here’s a tiny problem illustrating our limited knowledge of finite fields : “Imagine an infinite queue of Knights \{ K_1,K_2,K_3,\hdots \}, waiting to be seated at the unit-circular table. The master of ceremony (that is, you) must give Knights K_a and K_b a place at an odd root of unity, say \omega_a and \omega_b, such that the seat at the odd root of unity \omega_a \times \omega_b must be given to the Knight K_{a \otimes b}, where a \otimes b is the Nim-multiplication of a and b. Which place would you offer to Knight K_{16}, or Knight K_n, or, if you’re into ordinals, Knight K_{\omega}?”

What does this have to do with finite fields? Well, consider the simplest of all finite field \mathbb{F}_2 = \{ 0,1 \} and consider its algebraic closure \overline{\mathbb{F}_2}. Last year, we’ve run a series starting here, identifying the field \overline{\mathbb{F}_2}, following John H. Conway in ONAG, with the set of all ordinals smaller than \omega^{\omega^{\omega}}, given the Nim addition and multiplication. I know that ordinal numbers may be intimidating at first, so let’s just restrict to ordinary natural numbers for now. The Nim-addition of two numbers n \oplus m can be calculated by writing the numbers n and m in binary form and add them without carrying. For example, 9 \oplus 1 = 1001+1 = 1000 = 8. Nim-multiplication is slightly more complicated and is best expressed using the so-called Fermat-powers F_n = 2^{2^n}. We then demand that F_n \otimes m = F_n \times m whenever m &lt; F_n and F_n \otimes F_n = \frac{3}{2}F_n. Distributivity wrt. \oplus can then be used to calculate arbitrary Nim-products. For example, 8 \otimes 3 = (4 \otimes 2) \otimes (2 \oplus 1) = (4 \otimes 3) \oplus (4 \otimes 2) = 12 \oplus 8 = 4. Conway’s remarkable result asserts that the ordinal numbers, equipped with Nim addition and multiplication, form an algebraically closed field of characteristic two. The closure \overline{\mathbb{F}_2} is identified with the subfield of all ordinals smaller than \omega^{\omega^{\omega}}. For those of you who don’t feel like going transfinite, the subfield ~(\mathbb{N},\oplus,\otimes) is identified with the quadratic closure of \mathbb{F}_2.

The connection between \overline{\mathbb{F}_2} and the odd roots of unity has been advocated by Alain Connes in his talk before a general public at the IHES : “L’ange de la géométrie, le diable de l’algèbre et le corps à un élément” (the angel of geometry, the devil of algebra and the field with one element). He describes its content briefly in this YouTube-video

At first it was unclear to me which ‘coupling-problem’ Alain meant, but this has been clarified in his paper together with Caterina Consani Characteristic one, entropy and the absolute point. The non-zero elements of \overline{\mathbb{F}_2} can be identified with the set of all odd roots of unity. For, if x is such a unit, it belongs to a finite subfield of the form \mathbb{F}_{2^n} for some n, and, as the group of units of any finite field is cyclic, x is an element of order 2^n-1. Hence, \mathbb{F}_{2^n}- \{ 0 \} can be identified with the set of 2^n-1-roots of unity, with e^{2 \pi i/n} corresponding to a generator of the unit-group. So, all elements of \overline{\mathbb{F}_2} correspond to an odd root of unity. The observation that we get indeed all odd roots of unity may take you a couple of seconds1.

Assuming we succeed in fixing a one-to-one correspondence between the non-zero elements of \overline{\mathbb{F}_2} and the odd roots of unity \mu_{odd} respecting multiplication, how can we recover the addition on \overline{\mathbb{F}_2}? Well, here’s Alain’s coupling function, he ties up an element x of the algebraic closure to the element s(x)=x+1 (and as we are in characteristic two, this is an involution, so also the element tied up to x+1 is s(x+1)=(x+1)+1=x. The clue being that multiplication together with the coupling map s allows us to compute any sum of two elements as x+y=x \times s(\frac{y}{x}) = x \times (\frac{y}{x}+1). For example, all information about the finite field \mathbb{F}_{2^4} is encoded in this identification with the 15-th roots of unity, together with the pairing s depicted as

Okay, we now have two identifications of the algebraic closure \overline{\mathbb{F}_2} : the smaller ordinals equipped with Nim addition and Nim multiplication and the odd roots of unity with complex-multiplication and the Connes-coupling s. The question we started from asks for a general recipe to identify these two approaches.

To those of you who are convinced that finite fields (LOL, even characteristic two!) are objects far too trivial to bother thinking about : as far as I know, NOBODY knows how to do this explicitly, even restricting the ordinals to merely the natural numbers!

Please feel challenged! To get you started, I’ll show you how to place the first 15 Knights and give you a procedure (though far from explicit) to continue. Here’s the Nim-picture compatible with that above

To verify this, and to illustrate the general strategy, I’d better hand you the Nim-tables of the first 16 numbers. Here they are

It is known that the finite subfields of ~(\mathbb{N},\oplus,\otimes) are precisely the sets of numbers smaller than the Fermat-powers F_n. So, the first one is all numbers smaller than F_1=4 (check!). The smallest generator of the multiplicative group (of order 3) is 2, so we take this to correspond to the unit-root e^{2 \pi i/3}. The next subfield are all numbers smaller than F_2 = 16 and its multiplicative group has order 15. Now, choose the smallest integer k which generates this group, compatible with the condition that k^{\otimes 5}=2. Verify that this number is 4 and that this forces the identification and coupling given above.

The next finite subfield would consist of all natural numbers smaller than F_3=256. Hence, in this field we are looking for the smallest number k generating the multiplicative group of order 255 satisfying the extra condition that k^{\otimes 17}=4 which would fix an identification at that level. Then, the next level would be all numbers smaller than F_4=65536 and again we would like to find the smallest number generating the multiplicative group and such that the appropriate power is equal to the aforementioned k, etc. etc.

Can you give explicit (even inductive) formulae to achieve this? I guess even the problem of placing Knight 16 will give you a couple of hours to think about… (to be continued).

  1. If m is odd, then (2,m)=1 and so 2 is a unit in the finite cyclic group ~(\mathbb{Z}/m\mathbb{Z})^* whence 2^n = 1 (mod~m), so the m-roots of unity lie within those of order 2^n-1
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