%% alf@mpce.mq.edu.au

%% Copyright 1993
%% Alf van der Poorten
%% ceNTRe for Number Theory Research
%% Macquarie University
%% NSW 2109 AUSTRALIA


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 \rightline{\it Der Fermatsche Satz ist zwar mehr
ein Curiosum}\rightline{\it  als ein Hauptpunkt der
Wissenschaft.}\smallskip
\rightline{E. E. Kummer\hskip1cm}\bigskip
\def\norm{\roman N}

Lam\'e's argument produced a flurry of investigation in Paris on the matter of
unique factorisation of cyclotomic integers. To understand the underlying
ideas we
had better recall just why the usual integers $\Z$ are known to have unique
factorisation.

The argument goes back to Euclid and relies on observing that given integers $a$
and $b$, with $b\ne0$, there is some multiple, say $kb$, of $b$ so that
$$|a-kb|\lt|b|\,.$$
Of course one picks $k$ as the integer part of the quotient of $a/b$; then
$a-kb$
is the {\it remainder\/}. The point is that a common divisor of
$a$ and
$b$ is also a common divisor of $b$ and of $r=|a-kb|$. Now iterating the
argument
reveals the greatest common divisor $d=(a,b)$ of
$a$ and
$b$ --- without one having to
factorise $a$ or $b$. 

In fact the process is the same thing as expanding $a/b$ as a {\it continued
fraction\/}:

$$\cfrac kn$$

usually abbreviated by writing $a/b=\CF{k_0\\k_1\\\ldots\\k_{n-1}\\k_n}$. Its
truncations
$$\CF{k_0\\k_1\\\ldots\\k_{h-1}\\k_h}=p_h/q_h$$
yield rational numbers (called {\it convergents\/} because they converge rapidly
to $a/b$, each approximating it surprisingly well) where $p_h$ and $q_h$ are
relatively prime, and are given by the recursive formul\ae\
$$\pmatrix p_h&p_{h-1}\\q_h&q_{h-1}\endpmatrix\pmatrix
k_{h+1}&1\\1&0\endpmatrix=
\pmatrix p_{h+1}&p_h\\q_{h+1}&q_h\endpmatrix,\;\hbox{ where\ }\quad
\pmatrix p_{-1}&p_{-2}\\q_{-1}&q_{-2}\endpmatrix=\pmatrix 1&0\\0&1\endpmatrix.$$
It follows that $p_hq_{h-1}-p_{h-1}q_h=(-1)^{h+1}$, and since by definition
$p_n/q_n=a/b$, we must have $a=dp_n$ and $b=dq_n$, where $d=\gcd(a,b)$. Hence
$$dp_nq_{n-1}-p_{n-1}dq_n=(-1)^{n+1}d=q_{n-1}a-p_{n-1}b$$
displays the gcd as a $\Z-linear$ combination $d=sa+tb$ of $a$ and $b$.


Since the
Euclidean Algorithm  itself relies on the {\it Well Ordering Principle\/}:

A non-empty set of positive integers contains a smallest element,

we might have appealed directly to that principle --- which is of course
equivalent to the Principle of Induction, to remark that the set 
$$I=(a,b)=\{sa+tb:s,t\in\Z\}$$
of all $\Z$-linear combinations of $a$ and $b$ has a least positive element $d$,
and that $d$ is fairly readily seen to be a greatest common divisor,
usually also
denoted by $(a,b)$, of $a$ and $b$.

It is easy to see, say by induction, that every positive integer is a product of
positive irreducible integers (one views $1$ as being an empty such
product). To see that the decompositions are unique, up to the order of the
factors, suppose to the contrary that $n$ has two essentially distinct
decompositions. Of course then $n$ is neither irreducible nor $1$, so we may
set
$n=ab$ with positive $a$ and $b$ both less than $n$. Our hypothesis now becomes
that there is some irreducible $q$ which divides $n=ab$ --- one writes
$q\div ab$
--- but neither $q\div a$ nor $q\div b$.

However, since $q$ is irreducible, $q\ndiv a$ entails that the gcd
$(a,q)=1$. Thus
there are integers $s$ and $t$, say, so that
$1=sa+tq$, whence $b=sab+tqb$, and evidently $q\div b$. Thus every
irreducible integer is {\it prime\/}, a prime being an element $p$, which
is {\it
not\/} a {\it unit\/} --- that is, $p\ndiv1$ --- with the property that
$p\div ab$
implies
$p\div a$ or $p\div b$.

Once every irreducible is prime, it follows immediately that decomposition into
irreducibles is unique.

Now consider, for example, the Eisenstein integers $\{a+\rho b:
a,b\in\Z\}$, where
$\rho=\frac12(-1+\sqrt{-3})$ so $\rho^3=1$. We associate with each such
number its
{\it norm\/}
$$\norm(a+\rho b)=(a+\rho b)(a+\rho^2 b)=a^2-ab+b^2\,.$$
One then shows, following the idea of division with remainder, that given
integers
$a+\rho b$ and $c+\rho d$,  there are integers $u+\rho v$ and $r+\rho s$ so that
$$a+\rho b=(u+\rho v)(c+\rho d)+(r+\rho s) \hbox{ with\ } \norm(r+\rho
s)\le\tfrac34\norm(c+\rho d)\,.$$
Thus the Eisenstein integers form a Euclidean ring with respect to the
norm, and,
just as for $\Z$, it follows that they constitute a unique factorisation domain.

In similar spirit one sees that the Gaussian integers $\{a+i b: a,b\in\Z\}$ have
the Euclidean property with respect to their norm $N(a+ib)=a^2+b^2$.

With rather more effort, Cauchy succeeded in guessing correctly that the domains
$\Z[\zeta_m]$ of cyclotomic integers are Euclidean for
$$m=5, 6, 7, 8, 9, 10, 12, 14, 15$$
as well; here $\zeta_m^m=1$, and the $\zeta_m$ are {\it primitive\/} $m$-th
roots
--- no smaller power of $\zeta_m$ is $1$. Mind you, this was a less
comprehensive
achievement than may appear. Cauchy appears to miss the fact that, when $n$
is odd, $\zeta_n=-\zeta_{2n}$, so $\Z[\zeta_n]=\Z[\zeta_{2n}]$. In addition to
these results, Cauchy also claimed to have analytic arguments suggesting that
$\Z[\zeta_m]$ is Euclidean for all large $m$, say bigger than $10$, but he was
unable to find a decisive proof.

The explanation for this failure came from Germany, in a letter from
Kummer to Liouville. At this time German mathematics was far more sophisticated 
than that in France, inspired by stars like Gauss, Jacobi and Dirichlet. On
April
28, 1847 Kummer explained that it is not the case in general that the domains
$\Z[\zeta_m]$ have unique factorisation, though he had succeeded in retrieving
unique factorisation by introducing {\it ideal\/} numbers.

Suitably chastened, Cauchy showed in the next {\it Comptes Rendus de
l'Acad\'emie
des Sciences\/} that, indeed, the numbers $\Z[\zeta_{23}]$ do not have the
property
of unique factorisation into irreducibles.

Once forewarned, it is easy to see that unique factorisation is in fact rather
unusual. Take for example the domain
$\Z[\sqrt{-5}]=\{a+\sqrt{-5}b:a,b\in\Z\}$. We
see that
$$6=2\cdot3=(1+\sqrt{-5})(1-\sqrt{-5})\,.$$
Plainly the irreducibles $2$ and $3$ are not prime; nor are the irreducibles
$1\pm\sqrt{-5}$. Kummer deals with this by introducing {\it ideal\/}
numbers --- the
word `imaginary' is already in use --- and shows that he may write, say,  
$2=\frak a\overline{\strut\frak a}$, $3=\frak b\overline{\strut\frak b}$,
whereby the
preceding decompositions become
$$6=\frak a\overline{\strut\frak a}\cdot\frak b\overline{\strut\frak b}=
\frak a\overline{\strut\frak b}\cdot\overline{\strut\frak a}\frak b.$$
The new ideal irreducibles are prime. At little more than the cost of writing a
few gothic letters, unique factorisation is restored.

These ideas had not arisen in the context of Fermat's Last Theorem, but rather
from a desire to generalise the {\it law of quadratic reciprocity
\/} of Gauss (1801). The law states that if $p$ and $q$ are distinct odd primes
then the two congruences
$$x^2\equiv p\mod q \quad\hbox{ and\ }\quad y^2\equiv q\mod p$$
either both have a solution in integers $x$ and $y$, or are both insoluble,
except
when both $p$ and $q$ are $3$ modulo $4$, in which case one is solvable and the
other is not. The question was to generalise this rule to powers higher than the
second.

By the way, at first glance such a reciprocity law doesn't seem a big deal. But
imagine that it is important to describe all primes $q$ such that $x^2\equiv p\mod
q$ has a solution. In principle, one must test each $q$, trying $\fl {\frac12q}$
different potential $x\mod q$. On the other hand, given quadratic
reciprocity, it
is immediately clear that the good $q$ are those of certain congruence classes
$\mod 4p$. This turns an apparently infinite problem into a finite one.

Gauss himself had shown in 1832 that in order to formulate a reciprocity law for
fourth powers one needed the Gaussian integers $\Z[i]$ and he had established a
cubic reciprocity law using the number $\Z[\rho]$. His results suggested
that for
higher powers $n$ one must first deal with the following question:

Can every prime $p$ congruent to $1$ modulo $n$ be written as the norm of a
cyclotomic integer in $\Z[\zeta_n]\,$?

Incidentally, the norm of
$a(\zeta_n)=a_0+a_1\zeta_n+\cdots+a_{n-1}\zeta_n^{n-1}$
is the product of the different $a(\zeta^r_n)$ as $\zeta_n^r$ runs through the
primitive $n$-th roots of unity. [It is not too hard to see that
$\zeta^r_n$ is a
primitive $n$-th root exactly when $r$ and $n$ are relatively prime. The
number of
such $r$ is denoted by $\phi(n)$, where $\phi$ is the so-called Euler totient
function.]

It seems clear that Jacobi, and later Eisenstein, realised that the
question just
posed has an affirmative answer if and only if the domain $\Z[\zeta_n]$ has
unique
factorisation. A beautiful proof of this is given by Kummer ({\it Collected
Papers\/}, Springer 1975, Vol I, 241--243). Apparently Kummer had suggested in
1844, in a paper withdrawn before publication, that the answer is yes for
all $n$,
but had made the timely discovery that this is false for $n=23$.

The mythology of Fermat's Last Theorem claims that Kummer perpetrated Lam\'e's
error, and that battered, but not defeated, he retrieved matters by
his introduction of ideal numbers. Whatever, it was in 1847, just when the
Parisian mathematicians were beating their heads against the wall of unique
factorisation, that Kummer got the idea of applying his ideal theory to
Fermat's Last Theorem.\footnote""{This section is substantially plagiarised from
the introduction to the thesis of Hendrik Lenstra Jr., which  I
translated from the Dutch for {\it Math. Intelligencer\/} {\bf 2} (1980), 6--15;
73--77; 99--103.}

As regards unique factorisation into irreducibles, it turns out that there
are just
$30$ different cyclotomic fields $\Q[\zeta_n]$ with unique factorisation: 
$$\multline
n=1,3,4,5,7,8,9,11,12,13,15,16,17,19,20,21,24,25,27,28,\\32,33,35,36,
40,44,45,48,60,84.\endmultline$$
The notion `Euclidean'
plays no role in the eventual argument [J. H. Masley and H. L. Montgomery,
`Cyclotomic fields with unique factorisation', {\it J. f\"ur Math.\/} {\bf
286/87}
(1976), 248--256.]. Showing that these fields have unique factorisation is
routine;
excluding all other cases is not quite. In 1979, only
$13$ of the fields were known to be Euclidean.

Kummer eventually established the higher reciprocity laws. It was a task he
valued
far more highly than his researches on Fermat's Last Theorem to which owes his
fame.
\medskip
{\it Bei meinen Untersuchungen \"uber die Theorie der complexen Zahlen und den
Anwendungen derselben auf den Beweis des Fermatschen Lehrsatzes, welchen ich der
Akademie der Wissenschaften vor drei jahre mitzutheilen die Ehre gehabt
habe, ist
es mir gelungen die allgemeinen Reciprocit\"atsgesetze f\"ur beliebig hohe
Potenz\-reste zu entdecken, welche nach dem gegenw\"artigen Stande der
Zahlentheorie
als die Hauptaufgabe und die Spitze dieser Wissenschaft anzusehen
sind.\/}\smallskip
\rightline{E. E. Kummer (1850)\hskip1.5cm}


 
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