of Niels Henrik Abel

44 Pages·2008·18.74 MB·English

Olav Arnfinn Laudal- Ragni Piene (Editors)
The Legacy
of Niels Henrik Abel
The Abel Bicentennial,
Oslo, 2002
Springer
Editors:
Olav Arnfinn Laudal
Ragni Piene
,
University of Oslo
Department of Mathematics
0316 Oslo, Norway
e-mail: [email protected]
URL: http://folk.uio.no/arnfinnll
http://folk. uio.no/ragnip/
I
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Mathematics Subject Classification (2000): Ol-XX, ll-XX, 14-XX, 16-XX, 32-XX, 34-XX, 37-XX, 51-XX
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Contents
The Legacy of Niels Henrik Abel .............. ........... .. .... . .
King Harald V
Opening address
The Abel Bicentennial Conference University of Oslo, June 3, 2002 . . . . . 5
Arild Stubhaug
The Life of Niels Henrik Abel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Christian HouzeI
The Work of Niels Henrik Abel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 21
PhiIlip Griffiths
The Legacy of Abel in Algebraic Geometry ........ .. ........... ... 179
Daniel Lazard
Solving Quintics by Radicals ... .... .... ... .. ... ..... .... .... .. .. 207
Birgit Petri and Norbert Schappacher
From Abel to Kronecker: Episodes from 19th Century Algebra ......... 227
Giinther Frei
On the History of the Artin Reciprocity Law in Abelian Extensions
of Algebraic Number Fields: How Artin was Led to his Reciprocity Law . 267
Aldo Brigaglia, Ciro Cmberto, Claudio Pedrini
The Italian School of Algebraic Geometry and Abel's Legacy .. ...... .. 295
Fabrizio Catanese
From Abel's Heritage: Transcendental Objects
in Algebraic Geometry and Their Algebraization ... .. . . . . . . . . . . . . . .. 349
Steven L. Kleiman
What is Abel's Theorem Anyway? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 395
Torsten Ekedahl
On Abel's Hyperelliptic Curves ..... .......... ................... 441
VI Contents
Mark Green, Phillip Griffiths
Formal Deformation of Chow Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 467
Herbert Clemens
An Analogue of Abel's Theorem ................................. 511
Tom Graber, Joe Harris, Barry Mazur, Jason Starr
Arithmetic Questions Related to Rationally Connected Varieties ... ..... 531
Yum-Tong Siu
Hyperbolicity in Complex Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 543
G.M.Henkin
Abel-Radon Transform and Applications ... ... . . . . . . . . . . . . . . . . . . .. 567
Simon Gindikin
Abel Transform and Integral Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . .. 585
v.
P. Palamodov
Abel's Inverse Problem and Inverse Scattering .. . . . . . . . . . . . . . . . . . . .. 597
Jan-Erik Bjork
Residues and i)-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 605
Mikael Passare, August Tsikh
Algebraic Equations and Hypergeometric Series. . . . . . . . . . . . . . . . . . . .. 653
mlkan Hedenmalm
Dirichlet Series and Functional Analysis. . . . . . . . . . . . . . . . . . . . . . . . . .. 673
Yuri I. Manin
Real Multiplication and Noncommutative Geometry .... ........... .. 685
William FuIton
On the Quantum Cohomology of Homogeneous Varieties . . . . . . . . . . . .. 729
Christian Kassel
Quantum Principal Bundles up to Homotopy Equivalence . . . . . . . . . . . .. 737
Michel van den Bergh
Non-commutative Crepant Resolutions .. ... .. .......... ...... .. ... 749
Moira Chas, Dennis Sullivan
Closed String Operators in Topology Leading to Lie Bialgebras
and Higher String Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 771
From Abel to Kronecker:
Episodes from 19th Century Algebra
Birgit Petri and Norbert Schappacher
Recalling Niels Henrik Abel on the Resolution of Algebraic Equations
2 1853-1856: The General Form of Solvable and.C yclic Equations
3 1858-1861: Elliptic Functions and the Quintic Equation
4 1861-1921: Leopold Kronecke.r vs. Felix Klein
5 Glimpses Beyond
References
This paper is about Leopold Kronecker reading Niels Henrik Abel's results and
ideas on the resolution of algebraic equations. When the young Kronecker began to
work on algebraic equations, he went back and forth between Abel's works and his
own ideas. And throughout his career he continued to position himself much nearer to
Abel than to Galois. At the same time, his own creativity transformed Abel's results
and questions into something more arithmetic and fairly different. For instance,
already in his very first publication on algebraic equations [41], when unfolding
Abet's problems on solvable equations, Kronecker essentially claimed both what
is known today as the 'Theorem of Kronecker and Weber,' to the effect that every
abelian extension of Q is cyclotornic, and its analogue for abelian extensions of
QC.J=l), and even indicated further generalizations.
Our article was written with two different kinds of readers in mind: interested
mathematicians as well as historians of mathematics. The mathematician, espe
cially the number theorist, may appreciate for example our attempt at partially
reconstructing Kronecker's reasoning for the priine-degree-case of the 'Kronecker
Weber-Theorem'. The historian will notice the newly used documents. For instance,
we draw on, and publish here for the first time, a few letters from Kronecker to
Dirichlet, of which Harold M. Edwards has made us aware and shared his personal
transcription with us. We also make use of some of the unpublished, handwrit
ten notes of Kronecker's lecture courses which are preserved in the library of the
Strasbourg mathematical institute IRMA.
Both the mathematical and the historical reader, however, will not fail to realize
the peculiar position that the mathematical events discussed here occupy with respect
to what can be considered (at least with hindsight) as the mainstream development
228 B. Petri and N. Schappacher
of Galois Theory in the nineteenth century. Comparing for instance Kronecker's first
paper in the subject [41J to Enrico Betti's big memoir [9], which had appeared only
a year before, and which Kronecker knew, the two texts almost seem to belong to
different mathematical cultures: Betti's treatise is justly regarded to be a milestone
in the development of Galois Theory in that it treats permutations first - if in a way
which is still quite far from our group theory, and actually quite hard to penetrate for
mathematician and historian alike -, and their applications to the theory of algebraic
equations in a separate part thereafter. Kronecker's Berlin Academy Note on the
other hand barely sketches proofs, and takes a dramatic number theoretic turn at the
end, which may well have been the very starting point of his work.
The final Sect. 4 of our paper concentrates on elements of the history of Kro
necker's 1861 sharpening of Abel's theorem on the non-resolubility of the gen
eral quintic equation: Kronecker saw that such an equation does not admit one
parameter resolvents. This was preceded by several proposals - due to Hermite,
Kronecker, and Brioschi - to use elliptic functions in the resolution of the quin
tic equation; we will describe them briefly in Sect. 3. On the other hand, Kro
necker's theorem of 1861 gave rise to a lasting difference of opinions between
Kronecker and Felix Klein about the limits of algebraic resolutions of the quin
tic equation. It was Klein who supplied the first published proof of Kroneck
er's result; but we will try to explain the difference of his point of view from
Kronecker's.
The present paper thus highlights a few seminal episodes from nineteenth century
algebra, which are directly linked to the name of Abel, and which, even though partly
forgotten, are part of our mathematical heritage.
1 Recalling Niels Henrik Abel on the Resolution
of Algebraic Equations
1.1. Since early mathematics in several different cultures had some knowledge of
solving what we consider today as special cases of quadratic polynomial equations,
it is probably impossible to pin down a precise first occurence of the general for
mula for the solutions of all quadratic equations x2 + px + q = 0, i.e., of the
formula
X = --p2 + -21( -1) K y ~L1 (for K = 0, 1),
K
where L1 = p2 - 4q, and where the square root is fixed in some way.
1.2. The resolution of cubic equations was largely accomplished by Tartaglia and
Cardano around 1540. In today's notation, if x3 + px + q = 0 (a case to which one
is easily reduced by what was later to be called a Tschirnhausen transformation),
From Abel to Kronecker 22l}
and if we set .1 = -4 p3 - 27 q2, then, for a suitable choice of the two cubic roots,
one obtains the solutions:
(forK = 0,1,2),
where Q is a primitive third root of 1. I
1.3. Skipping the case of degree four (in fact, we will only consider equa
tions of prime degree further on), the next bit of general mathematical cul
ture in the theory of algebraic equations is of course the impossibility of re
solving the general quintic equation by radicals, which was proved com
pletely for the first time by Abel - see [2]. This fact is usually linked today
to the simplicity of the altemating group on 5 letters, even though Abel's orig
inal proof was not group-theoretical in nature, but proceeded, in a fashion very
typical of Abe!, via an analysis of the most general form of an algebraic ex
pression involving radicals which could possibly be a solution to the quintic
equation.
In order to prove the impossibility rigorously, Abel carefully spelled out, what
'resolution by radicals' was supposed to mean. As a consequence he proved in
particular:2
If an equation is algebraically solvable, then the root can always be written in such
a way that all algebraic functions of which it is composed can be expressed by
rational functions of the roots of the given equation.
In modem words, this means that the constructions of the resolution by radicals
are realized within the splitting field of the given polynomial. This statement was
tumed by Kronecker in 1861 into a criterion that every notion of algebraic solvability
of an algebraic equation should meet. He called it AbeI's postulate, and we shall
see in Sect. 4 below that it played an important role in his strengthening of Abel's
impossibility theorem. 3
1 Let us note in passing that Eisenstein published a short note [16] where he tried to write
down the fOlIDulae for equations of degree 1,2,3, and 4 in a coherent way, and where he
added a cryptic footnote indicating an analytic resolution of the quintic equation - see [61].
2 See the indented passsage at the end of §II of [2]. Kronecker quoted this in [46, p. 55-56]
from the German translation of Abel's paper as it appeared in Crelle's journal: Journal
fiir die reine uoo angewaOOte Mathematik 1 (1826), p. 73, tacitly adjusting the spelling
in two places: Wenn eine Gleichung algebraisch aufiosbar ist, so kann man der Wurzel
allezeil ein[eJ solche Form geben, daj3 sich alle algebraische[nJ Functionen, aus welchen
sie zusammengesetzt ist, durch rationale Functionen del' Wurzeln der gegebenen Gleichung
ausdriicken lassen.
3 In his lecture course [52, p. 164], Kronecker introduced this postulate with these words: Eine
Gleichung aufiosen heiJ3t sie ersetzen durch eine Kette von Gleichungen vorgeschriebener
Beschaffenheit; das Wesen der "Kette" besteht darin, dass immer aus einer Gleichung
eine GrojJe for die folgenden bestimmt wird. Bei der Aufiosung durch reine Gleichungen
230 B. Petri and N. Schappacher
1.4. It is less well-known that Abel also investigated, around the same time, the
general shape ofthe roots of an equation of degree 5 (or higher) which does happen
to be solvable by radicals. On 14 March 1826, he explained in a letter to Crelle4 that,
if a quintic equation with rational coefficients is solvable by radicals, then
x = c + A . a 51 . a2l5 . 4a25 .3 a35 + A I . al15 . a225 . 4a35 .3 a 5 +
+A 2 . a21') . a235 . 4a ') .3 al5 + A 3 . a351 . a2 ') . 4al" .3 a25 '
!
where a = m + n~ + h (1 + e2 + ~) ,
=m -n~+!h(l +e2-~),
al
a2 =m+n~ -Jh(l+e2+~),
a3 =m -n~-Jh(l+e2-~),
and A = K + K'a + K"a2 + K"'aa2, Al = K + K'al + K"a3 + K"'ala3,
A2 = K + K'a2 + K"a + K"'aa2, A3 = K + K'a3 + K"al + Klllala3,
with c, h, e, m, n, K, K', K", K"' rational numbers. Abel also claimed he had anal
ogous formulce in degree "7, 11, 13, etc."
1.5. Abel came back to this problem of writing down the general form of the roots of
solvable equations shortly before his death in a manuscript [7] of which only the first
pages are fully worked out, and which ends in a sequence of unconnected formulas.
One of the claims in this manuscript is that the root x of any solvable equation of
prime degree f.L with rational coefficients takes the form:
x = A + \fij; + ~ + ... + <!RfL- I ,
with rational A and where RI, ... , RfL- I are roots of an equation of degree f.L - 1.
gilt nun der von Abel entdeckte Satz, - auJ dem Abels Unmoglichkeitsbeweis wesentlich
beruht -, dass alIe successive eingejiihrten Groj3en rationale Funktionen der Wurzeln
der auJzulOsenden Gleichung sind. Die Forderung, dass beijeder Aufiosung die successive
auftretenden Groj3en rationale Funktionen der Wurzeln der auJzulOsenden Gleichung seien,
solI als das "Abel'sche Postulat" bezeichnet werden. Die Berechtigung desselben liegt
darin, daj3, sowie man das Postulat auJgiebt, Gleichung derselben Gattung verschiedene
Aufiosungsmethoden notig haben, dass also bei der Vermittlung durch irrationale Funk
tionen die Einteilung der algebraischen Funktionen in Gattungen durchbrochen wird. Das
Abelsche Postulat ist daher identisch mit der Forderung, dass man keine Methoden der
Aufiosung algebraischer Gleichungen einjiihren dart. welche Gleichungen die derselben
Gattung angehoren auseinander reij3en. - The tenn Gattung here corresponds more or less
to the splitting field of a polynomial.
4 See Journaljiir die reine und angewandte Mathematik 5 (1830), p. 336; the French version
is reproduced in [1, vol. 2, p. 266].
From Abel to Kronecker 231
2 1853-1856: The General Form of Solvable
and Cyclic Equations
It was in particular Abel's result recalled in l.5 which the twenty-nine year old
Leopold Kronecker (1823-1891) had apparently discovered himself before he re
ceived, on the evening of30 January 1853, his copy of Abel's Collected Papers where
he could read [7] himself. Kronecker received Abet's Collected Works at Liegnitz
(at the time Lower Silesia, today Legnica, Poland), where he worked overseeing the
family estate and his uncle's banking business, trying at the same time - as much as
s
recurrent health problems would permit - to get actively back into mathematics.
In order to appreciate Abel's and Kronecker's points of view, one should compare
them to today's standard practice of teaching the Galois Theory of solvable equations
by showing that an equation can be solved by radicals if and only if its Galois group
is solvable. Such a presentation - if not its modem formulation in terms of field
extensions, automorphisms, etc. - is relatively close in spirit to Galois' original point
of view. That point of view actually took quite some time in the nineteenth century
to be understood and further developed. For instance, the criterion for solvability
which was especially popular in the middle of the nineteenth century was not group
theoretical, but rather the statement that "all roots of the equation can be expressed
rationally in terms of two of them.,,6
2.1 Principles. Abel's and Kronecker's approach to solvable equations was of a dif
ferent nature. As for Abel, one may notice throughout his works a certain tendency
to completely classify classes of mathematical objects by a fitting formalism. His
systematic treatment of elliptic integrals [4] can be seen as an example of this, which
the famous memoir on "abelian functions" sent to the Paris Academy [3] carries
even further'? And in the long opening of the unfinished manuscript that Kronecker
was finally able to read in 1853, Abel presented his method of asking only questions
which specify the formal type of acceptable answers as "the only scientific" one,
and proudly recalled8:
I have treated several branches of analysis in this way, and even if I often took
up problems that were beyond my forces, I did obtain a lot of general results
5 See Kronecker's letter to Dirichlet of 31 January 1853 reproduced in Appendix I below. -
The edition of Abel's works that Kronecker read was of course the first edition, edited by
Holmboe in 1839. In the 1870's, Kronecker, together with Clebsch and Weierstrass, would
be among the first mathematicians to plead for a new edition of Abel's Works, which was
then realized by Sylow and Lie in 1881 - see the preface of [I].
6 Cf. [19], in particular p. 69.
7 Abel even tried this approach (unsuccessfully) on Fermat's Last Theorem - see [Abel
(Euvres, vo!. n, p. 254-255]
8 [7, p. 218]: J'ai traiUi plusieurs branches de ['analyse de cette maniere, et quoique je
me sois souvent propose des problimes qui ant depasse mes forces, je suis neansmoins
a
parvenu un grand nombre de resultats generaux quijettent un grand jour sur la nature
des quantites dont la connaissance est I' obj et des mathematiques.
232 B. Petri and N. Schappacher
which elucidate the nature of the quantities the knowledge of which is the object of
mathematics.
Abel had already applied his general principles to solvable algebraic equations once,
by singling out those which (in nowaday's parlance) have an abelian Galois group
see [6]. In the fragment [7], however, he was led to the following two questions
concerning all solvable algebraic equations9:
1. Find all equations of a given degree which are algebraically solvable.
2. Determine whether or not a given equation is algebraically solvable.
And he announced that his investigation would produce10
several general propositions about the sol vability of equations and about the nature of
their roots. It is these general properties that really make up the theory of algebraically
solvable equations, because it is of little importance to know whether a particular
equation is solvable or not. One of these general properties is for example that it is
impossible to solve algebraically the general equations of degree higher than four.
This plea for a theory yielding the general form of the roots of all solvable equations
(with rational coefficients) apparently squared perfectly with Kronecker's own ap
proach.lI He expressed his overall goals at many places, appealing in a very German
way to 'essence' (das Wesen) and the 'trne nature' of things. For instance, in the
beginning of the letter to Dirichlet of 31 January 1853 reproduced in Appendix I,
we read:
Having looked a bit more closely at the theory of solvable equations for a few weeks,
I have noticed a few things which seem to me to cast a great deal of light on the
essence of this matter.
And a little later in the same letter:
9 [7, p. 219]: 1. Trouver toutes les equations d'un degre derelmine queZconque qui soient
resolubZes algebriquement. 2. Juger si une equation donnee est resoluble algebriquement,
ou non.
a
10 [7, p. 219]: Dans le cours des recherches on parviendra plusieurs propositions generales
a a
sur les equations par rapport leur resolubilite et la forme des racines. C' est en ces
a
proprietes generales que consiste veritablement la theorie des equaliosn quant leur
resolution algebrique, car if importe peu si l'on sait qu'une equation d'une fO/me par
ticuliere est resoluble ou non. Une des ces proprieres generales est par exemple qu'it
est impossible de resoudre algebriquement les equations generales passe le quatrieme
degre. - This announces Abel's second proofof this famous impossiblility result, of which
Malmsten obtained a variant in [59].
11 According to Kronecker, the topic of solvable equations had been mentioned to him by
his formal thesis advisor Dirichlet already around 1843, as an easily accessible domain
of research - see the letter from Kronecker to Dirichlet dated 6 May 1853, reproduced in
Appendix I below.