In the simplest case where is a cyclic extension of degree and contains all -th roots of unity (and is coprime to the characteristic of ), this essentially means proving that if has cyclic Galois group of order , then there is some with and belongs to .

Indeed, the converse is relatively simple (in the technical sense that I can do it on paper or on the blackboard without having to think about it in advance, by just following the general principles that I remember).

I had however the memory that the second step is trickier, and didn’t remember exactly how it was done. The texts I use (the notes of M. Reid, Lang’s “Algebra” and Chambert-Loir’s delightful “Algèbre corporelle”, or rather its English translation) all give “the formula” for the element but they do not really motivate it. This is certainly rather quick, but since I can’t remember it, and yet I would like to motivate as much as possible all steps in this construction, I looked at the question a bit more carefully.

As it turns out, a judicious expansion and lengthening of the argument makes it (to me) more memorable and understandable.

The first step (which is standard and motivated by the converse) is to recognize that it is enough to find some element in such that , where is a generator of the Galois group and is a primitive -th root of unity in . This is a statement about the -linear *action* of on , or in other words about the representation of on the -vector space . So, as usual, the first question is to see what we know about this representation.

And we know quite a bit! Indeed, the *normal basis theorem* states that is isomorphic to the left-regular representation of on the vector space of -valued functions , which is given by

.

(It is more usual to use the group algebra , but both are isomorphic).

The desired equation implies (because is generated by ) that is a sub-representation of . In , we have an explicit decomposition in direct sum

where runs over all characters (these really run over all characters of over an algebraic closure of , because contains all -th roots of unity and has exponent ). So (if it is to exist) must correspond to some character. The only thing to check now is whether we can find one with the right eigenvalue.

So we just see what happens (or we remember that it works). For a character such that , and the element corresponding to under the -isomorphism , we obtain . But by easy character theory (recall that is cyclic of order ) we can find with , and we are done.

I noticed that Lang hides the formula in Hilbert’s Theorem 90: an element of norm in a cyclic extension, with a generator of the Galois group, is of the form for some non-zero ; this is applied to the -th root of unity in . The proof of Hilbert’s Theorem 90 uses something with the same flavor as the representation theory argument: Artin’s Lemma to the effect that the elements of are linearly independent as linear maps on . I haven’t completely elucidated the parallel however.

(P.S. Chambert-Loir’s blog has some recent very interesting posts on elementary Galois theory, which are highly recommended.)

]]>Algebra would look very different without her (“successive sets of symbols with the same second suffix“).

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LemmaLet be an odd prime number, let be an integer and let be a -tuple of elements of . For any subset of , denote

and for any , let

denote the multiplicity of among the .

Then if none of the is zero, there exists some for which isodd.

I will explain two proofs of this result, first Irving’s, and then one that I came up with. I’m tempted to guess that there is also a proof using some graph theory, but I didn’t succeed in crafting one yet.

**Irving’s proof.** This is very elegant. Let be a primitive -th root of unity. We proceed by contraposition, hence assume that all multiplicities are even. Now consider the element

of the cyclotomic field . By expanding and using the assumption we see that

In particular, the norm (from to ) of is an even integer, but because is odd, the norm of is known to be odd for all . Hence some factor must have , as desired.

**A second proof.** When I heard of Irving’s Lemma, I didn’t have his paper at hand (or internet), so I tried to come up with a proof. Here’s the one I found, which is a bit longer but maybe easier to find by trial and error.

First we note that

is even. In particular, since is odd, there is at least some with *even*.

Now we argue by induction on . For , the result is immediate: there are two potential sums and , and so if , there is some odd multiplicity.

Now assume that and that the result holds for all -tuples. Let be a -tuple, with no equal to zero, and which has all multiplicities even. We wish to derive a contradiction. For this, let . For any , we have

by counting separately those with sum which contain or not.

Now take such that is odd, which exists by induction. Our assumptions imply that is also odd. Then, iterating, we deduce that is odd for all integers . But the map is surjective onto , since is non-zero. Hence our assumption would imply that all multiplicities are *odd*, which we have seen is not the case… Hence we have a contradiction.

So the question is: who first proved the full “Peter-Weyl” Theorem for all compact groups? Pontryaguin, in 1936, certainly does, without remarking that Peter-Weyl didn’t, possibly because it was clear to anyone that the argument would work as soon as an invariant measure was known to exist. But since there are “easier” proofs of the existence of Haar measure for compact groups than the general one for all locally-compact groups (using some kind of fixed-point argument), it is not inconceivable that someone (e.g., von Neumann) might have made the connection before.

In fact, there is an amusing mystery in connection with Pontryaguin’s paper and von Neumann: concerning Haar measure, he refers to a paper of von Neumann entitled *Zum Haarschen Mass in topologischen Gruppen*, and gives the helpful reference *Compositio Math., Vol I, 1934*. So we should be able to read this paper on Numdam? But no! The first volume of Compositio Mathematica there is from 1935; it is identified as Volume I, and there is no paper of von Neumann to be found…

[**Update**: as many people pointed out, the paper of von Neumann is indeed on Numdam, but appeared in 1935; I was tricked by the absence of 1934 on the Compositio archive and the author's name being written J.V. Neumann (I had searched Numdam with "von Neumann" as author...)]

Here is what I came up with:

(i) *“The Elements of the Most Noble game of Whist; elucidated and discussed in all details”*, by A. Bandersnatch, Duke Dimitri, N. Fujisaki, A. Grothendieck, Y. Grünfiddler, J. Hardy, Jr., B. Kilpatrick and an Anonymous Person.

(ii) *“Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom funften Grade”*, by F. Klein; with barely legible annotations and initialed “HW” on the first page.

(iii) *“Histoire Naturelle”*, Volume XXXIII: Serpens, by George Louis Leclerc, Comte de Buffon, edition of 1798; initialed “A.K.” on the title page.

(iv) *“Chansons populaires de Corse, Navarre et Outre-Quiévrain”*, collected and commented by P. Lorenzini.

(v) “奥の細道” (*Oku no Hosomichi*), by Matsuo Bashò.

(vi) *“On the care of the pig”*, by R.T. Whiffle, KBE.

(vii) *“La Chartreuse de Parme”*, by Stendhal.

(viii) *“Der tsoyberbarg”*, by T. Mann; Yiddish translation by I.B. Singer of “Der Zauberberg”.

(ix) *“Les problèmes d’un problème”*, by P. Ménard; loose manuscript.

(x) Opera Omnia of L. Euler, volumes 1, 2, 7, 11, 13, 23, 24, 30, 56, 62, 64, 65 and 72.

(xi) *“Discorsi sopra la seconda deca di Tito Livio”*, by N. Macchiavelli.

(xii) *“An account of the recent excavations of the Metropolitan Museum at Khróuton, in the vicinity of Uqbar”*, by E.E. Bainville, OBE.

(xiii) Die Annalen der Physik, volumes 17, 18, 23 and 25.

(xiv) *“Diccionaro y gramática de la lengua Tehuelve”*, anonymous; attributed on the second page to “a Humble Jesuit of Rank”.

(xv) *“Le roi cigale”*, French translation by Jacques Mont–Hélène of an anonymous English romance.

(xvi) *“Mémoires du Général Joseph Léopold Sigisbert Hugo”*, by himself, with an Appendix containing the “Journal historique du blocus de Thionville en 1815, et des sièges de cette ville, Sierck et Rodemack en 1815″.

(xvii) *“Le Comte Ory”*, full orchestral score of the opera by G. Rossini with Libretto by E. Scribe and Charles-Gaspard Delestre-Poirson.

(xviii) *“Absalom, Absalom”*, by W. Faulkner; first edition, dedicated To R.C. on the second page.

(xix) *“Ficciones”*, by J-L. Borges, third edition with page 23 missing.

(xx) *“Die Gottardbahn in kommerzieller Beziehung”*, by G. Koller, W. Schmidlin, and G. Stoll.

(xxi) *“The etchings of the Master Rembrandt van Rijn, faithfully reproduced in the original size”*, anonymous.

(xxii) *“The memoirs of General S.I. Kemidov”*, by Himself.

(xxiii) *“Catalogue raisonné des œuvres d’Anton Fiddler”*, by W.B. Appel.

(xxiv) *“Zazie dans le métro”*, by R. Queneau.

(xxv) *“The mystery of the green Penguin”*, by E. Mount.

(xxvi) *“Χοηφόροι”* (The Libation Bearers), by Aeschylus; an edition printed in Amsterdam in 1648.

(xxvii) *“The Saga of Harald the Unconsoled”*, Anonymous, translated from the Old Norse by W.B. Appel.

(xxviii) *“Uncle Fred in the Springtime”*, by P.G. Wodehouse.

(xxix) *“The Tempest”*, by W. Shakespeare.

(xxx) *“The 1926 Zürich International Checkers Tournament, containing all games transcribed and annotated according to a new system”*, by S. Higgs.

(xxxi) *“A day at the Oval”*, by G.H. Hardy.

(xxxii) *“Harmonices Mundi”*, by J. Kepler, initialed I.N on the second page.

(xxxiii) *“Stories of cats and gulls”*, by G. Lagaffe.

(xxxiv) *“Traité sur la possibilité d’une monarchie générale en Italie”*, by N. Faria; loose handwritten manuscript on silk.

(xxxv) *“Broke Down Engine”*, 78 rpm LP record, interpreted by Blind Willie McTell.

(xxxvi) *“Les plages de France, Belgique et Hollande”*, by A. Unepierre.

(xxxvii) *“La Légende du Cochon Voleur et de l’Oiseau Rageur”*, traditional folktale, translated from the Arabic by P. Teilhard de Chardin.

(xxxviii) *“Discours des Girondins”*, collected and transcribed by a parliamentary committee under the auspices of the “Veuves de la révolution française”, published by Van-den-Broeck, Bruxelles in 1862.

(1) May 14 and 15, we will have the Number Theory Days 2015 at EPF Lausanne; the speakers are Gaetan Chenevier, Henryk Iwaniec, Alena Pirutka, Chris Skinner and Zhiwei Yun; this is co-organized by Ph. Michel and myself.

(2) Immediately afterward, from May 18 to 22, comes a conference at FIM, co-organized by H. Iwaniec, Ph. Michel and myself, with the title of “Analytic Aspects of Number Theory”; the current list of speakers is to be found on the web page; here is the poster (which is based on a picture taken by Henryk around Zürich last Fall):

Most importantly, there is a certain amount of funding available for local expenses of your researchers (doctoral and postdoctoral students). Applications can be made here (before Feburuary 6; the form states January 28, but this is an error that will be corrected).

]]>Proust s’est inspiré d’Henri Kowalski né en 1841, fils d’un officier polonais émigré en Bretagne. Il était à la fois compositeur de musique et concertiste.

or

Proust used as model Henri Kowalski, born in 1841, son of a Polish officer who emigrated to Brittany. He was a composer as well as a concert player.

to quote an authoritative list of Proust characters.

That Henri Kowalski is, it turns out, the son of Nepomus Adam Louis Kowalski, whose brother was Joachim Gabriel Kowalski, one of whose sons was Eugène Joseph Ange Kowalski, one of whose sons was Louis André Marie Joseph Kowalski, the fourth son of whom was my father. This puts me at genealogical distance (at most) six to a character from Proust. (Of course, rumors that Viradobetski was inspired by someone else can be safely discarded).

Wikipedia has a small page on Henri Kowalski, who was quite active and successful as a musician, and a rather impressive traveller. He spent thirteen years in Australia, leaving enough traces to be the subject of public lectures at the university of Melbourne. There are a few of his pieces on Youtube, for instance here. He also wrote a travel book which I now intend to read…

]]>In the first lecture, I mentioned a result of Fouvry as a motivation behind the study of other arithmetic functions in arithmetic progressions: roughly speaking, if one can prove that the exponent of distribution of the divisor functions ,…, is strictly larger than , then the same holds for the primes in arithmetic progressions.

This statement (which I will make more precise below, since there are issues of detail, including what type of distribution is implied) is very nice. But it turned out that quite a few people at the school were not aware of it before. The reason is probably to a large extent that, as of today (and as far as I know…), it has not been possible to use this mechanism to prove unconditional results about primes: the problem is that one does not know how to handle divisor functions beyond … One could in fact interpret this as saying that higher divisor functions are basically as hard as the von Mangoldt function when it comes to such questions.

The precise statement of Fouvry is Theorem 3 in his paper “Autour du théorème de Bombieri-Vinogradov” (Acta Mathematica, 1984). The notion of exponent of distribution of a function concerns a fixed residue class , and the average over moduli (with coprime to ) for some of the usual discrepancy

The actual assumptions concerning , , is a bit more than having this exponent of distribution : this must be true also for all convolutions

where is an arbitrary essentially bounded arithmetic function supported on a very short range for some .

This extra assumption is reasonable because since can be arbitrarily small, certainly all known methods to prove exponents of distribution larger than would accommodate this tweak.

As far as the proof is concerned, this Theorem 3 is actually rather “simple”: using the Heath-Brown identity, all the hard work is moved to the proof of an exponent of distribution beyond for the characteristic function of integers having no prime factors for and . This is much deeper, and involves all the machinery of dispersion and Kloostermania…

In addition, Fouvry mentioned to me the following facts, which I didn’t know, and which are very interesting from a historical point of view. First, this theorem of Fouvry is a strengthened version of the results of Chapter III of his *Thèse de Doctorat d’État* (Bordeaux, September 1981, supervised by J-M. Deshouillers and H. Iwaniec). At that time, Kloostermania was under construction and Fouvry had only Weil’s classical bound for Kloosterman sums at his disposal, and this original version required an exponent of distribution beyond for the functions . This illustrates the strength of Kloostermania!

Moreover, in this thesis, Fouvry used an iteration of Vaughan’s identity, instead of Heath-Brown’s identity, which only apparead in 1982. However, although this was less elegant, this iteration had the same property to transform a sum over primes into multilinear sums where all non smooth variables have small support near the origin.

Fouvry also suggests the following inverse challenge for *aficionados*: assuming an exponent of distribution for the sequence of primes, can one prove a similar exponent of distribution for all the divisor functions ?