Remarks on the History of Fermat’s Last Theorem 1844 to 1984

Michael Rosen, Brown University

Introduction .

It is arguably true that Fermat’s Last Theorem (FLT) has been the most famous of all mathematical problems for at least three centuries. There has been debate about whether it is a serious and important problem or merely a curiosity, but there can be no denying its popularity. Generations of mathematicians, both professional and amateur, have tried their hand at solving it. These efforts have resulted in a mighty body of theory with many deep and important results. Nevertheless, until 1984, when G. Frey connected the problem in an intimate way with the arithmetic theory of elliptic curves and the conjecture of Taniyama-Shimura-Weil (after earlier work in the same direction by Y. Helloguarch), a solution seemed as far away as ever.

We will attempt to review some of the highlights among the results obtained in the period from 1844 to 1984 (and a few that came after).

As is well known the study of Fermat’s last theorem for exponent n reduces rapidly to the cases where n = 4 or n is equal to an odd prime p. Fermat himself proved the theorem for n = 4 and claimed to have proven it for p = 3. By 1844 the theorem was known only for p = 3, 5, and 7 with proofs by L. Euler, L. Dirichlet and A. Legendre, and G. Lamé, respectively. The greatest contributions to our subject up to the last few years were made by E. Kummer. We begin our discussion with the year 1844, because that was the year Kummer published his results on the theory of ideal numbers in the field Q(ζ_{p}). There is some controversy over whether Kummer’s primary motivation for creating this theory was his interest in higher reciprocity laws or his interest in FLT. He was clearly interested in both although he thought the reciprocity laws were more important. In any case, three years later in 1847 he completed the proof of his great theorem: FLT is true for regular primes p, i.e. primes p which do not divide the class number of Q(ζ_{p}). To do this he had to define the class group, prove it is finite, analyze it as a product of two factors h_{p} = h_{p}^{+}h_{p}^{−}(the real class number times the relative class number), and do a profound investigation of the unit group. Beyond this he was able to relate the regularity of the prime p to divisibility by p of certain Bernoulli numbers. We will review this work and give a sketch of his proof of FLT for regular primes. Kummer claimed that FLT is true for regular primes p even if we allow entries from Q(ζ_{p}). Kummer’s proof of this latter claim contains an error, but his error was later patched up by Hilbert. It is interesting to ask if FLT is true for all odd primes p if we allow entries from Q(ζ_{p}). The modular proof over Z does not seem to extend easily to this case.

Kummer returned to FLT in 1857 when he published a paper dealing with certain cases of irregular primes p for which FLT can be be proven. There are some problems with this paper. The results were put on a firm foundation and further refined many years later (1929) by H.S.Vandiver. We will mention some of this work which is the foundation for calculations which have shown FLT to be true up to very large bounds. This culminated in a 1993 paper of J. Buhler-R. Crandall-R. Ernvall-T. Metsänkylä which showed FLT to be true for all primes up to four million. This may be of limited interest now that we know FLT is true without restriction, but these calculations have bearing on other issues as well, for example Vandiver’s conjecture that p does not divide h_{p}^{+} (already conjectured by Kummer many years earlier). The truth or falsity of this is still unknown, but it is true for all primes up to four million.

Irregular primes and Vandiver’s conjecture both relate to the interesting question of the structure of A_{p}, the p-primary part of the class group of Q(ζ_{p}). Kummer made a fundamental contribution to this by proving a special case of Stickelberger’s theorem on the prime decomposition of Gauss sums. We will discuss the consequences of this, among them the important theorem of J. Herbrand (1932). Its converse was proved by K. Ribet in 1976. Ribet’s proof used the arithmetic theory of modular forms and Galois representation theory and can thus be seen as an early breakthrough demonstrating the power of the methods which eventually led to a proof of FLT. We will also discuss subsequent work of A. Wiles and Mazur-Wiles on the structure of A_{p}.

The first case of FLT, i.e. when x^{p} + y^{p} = z^{p} and it is assumed that (xyz, p) = 1, will be dealt with elsewhere in this volume [Co]. We will concentrate on the second case. With a subject so vast it is inevitable that much of value will be left out. We hope that the material to be included will be sufficient to give a good sense of the fascinating and intricate history of this old conjecture which is finally a theorem.

Before beginning, we point out that we will assume the reader is familiar with elementary algebraic number theory and the elements of p-adic numbers. We will not attempt to use the language and notation of the nineteenth century, but will use more or less standard modern notation. What is lost in historical flavor by this process is made up for by an increase in clarity (or so we hope).

Section 1. FLT for Polynomials.

We begin our discussion, somewhat a-historically, by considering x^{n} +y^{n} = z^{n}, n > 2, over a polynomial ring A = k[T] where k is a field of characteristic prime to n. We claim there are no non-zero solutions in A except possibly for constant solutions, i.e. when x, y, and z are in k [Gr]. Suppose there is a solution x, y, z ∈ A where x, y, and z are non-zero and the maximum of the three degrees is d > 0. We will show that there is a solution x′, y′, and z′ all non-zero and with the maximum of the three degrees d′ satisfying d > d′ > 0. We could then repeat the process indefinitely. Of course, this is constitute a contradiction since there are only finitely many positive integers less than d. This is Fermat’s method of “infinite descent”.

Before we begin, note that it is no restriction to assume that k is algebraically closed. Further, since A is a unique factorization domain, a moment’s reflection shows that we may assume that x, y, and z are relatively prime. Now, consider the identity

∏_{i=0}^{n−1}(x + ζ^{i}y) = −z^{n}.  (1)

Here, ζ is a primitive n’th root of unity in k^{∗}. The factorization in equation (1) is the basis of all attempts to prove FLT before the elliptic curve/modular function approach. Every pair of factors on the left are relatively prime since if w divided the i’th and j’th factor it would also divide (ζ^{j}−ζ^{i})y and (ζ^{j−i}−1)x and these two polynomials are relatively prime. It follows easily that each factor is itself a constant times an n’th power. Since we have assumed that k is algebraically closed, each constant is also an n’th power. Thus, there are polynomials u,v, and w such that

x + y = u^{n}, x + ζy = v^{n}, x + ζ^{2}y = w^{n}.

Eliminating x and y from these three equations, we find

w^{n} + ζu^{n} = (1 + ζ)v^{n}.

Again using that k is algebraically closed we set x′ = w, y′ = ^{n} \sqrt{ζ} u, and z′ = ^{n} \sqrt{1 + ζ} v. Then, x′, y′, and z′ constitute a new solution with the required properties. In fact, d′ ≤ d/n.

It may be asked where the assumption n > 2 was used? The point is that if n = 2, then ζ = −1 and so 1 + ζ = 0 and z′ = 0. This shows that the proof breaks down if n = 2 but is sound for all n > 2.

Another interesting point is that the proof applies equally well to polynomials in several variables over a field.

Section 2. Kummer’s Work on Cyclotomic Fields.

The proof of FLT for the polynomial ring A = k[T] is very short and sweet. It contains the main ideas of the early attempts to prove Fermat’s assertion, but is much easier. Why is that? To begin with Z, the ring of rational integers, contains no roots of unity except ±1. To compensate for this the relevant roots of unity were added to Z and the arithmetic of the resulting rings investigated. Letting i = \sqrt{−1} C.F. Gauss investigated the ring Z[i] in his celebrated pair of papers on biquadratic reciprocity [G]. Let ω = e^{2πi/3}. The ring Z[ω] was investigated by C.G. Jacobi and independently by G. Eisenstein in the course of formulating and proving the law of cubic reciprocity. Gauss had investigated the same ring in an unpublished paper proving FLT for p = 3. These authors, and others, also experimented with other roots of unity. For small primes p it turns out that Z[ζ_{p}] is a unique factorization domain. In fact, this is true for all primes p less than 23. G.Lamé, perhaps led astray by this fact, announced in 1847 that he had a proof of FLT. Liouville almost immediately pointed out that he was implicitly assuming unique factorization. Soon thereafter, Liouville received a letter from Kummer which pointed out that not only is the assumption without proof, it is not correct. Kummer had shown three years earlier that Z[ζ_{23}] is not a unique factorization domain. As we shall see in a moment, Kummer had done much, much more than that.

There is a story, told by K. Hensel in an address given on the hundredth anniversary of Kummer’s birth, that Kummer himself had once constructed a proof of FLT assuming unique factorization and that the error had been pointed out to him by Dirichlet. Although this story has been widely retold in subsequent works on number theory, it is probably incorrect. This was pointed out, with fairly convincing evidence, by H. Edwards in a pair of papers [Ed2,Ed3] which are interesting reading both because of this issue and also for a general historical account of the events of 1844 to 1847 which bear on FLT.

Kummer not only noticed the failure of unique factorization, he invented his theory of ideal numbers to restore this immensely useful property for the rings Z[ζ_{p}]. Later, R. Dedekind extended Kummer’s work by discussing general rings of algebraic numbers and by reinterpreting Kummer’s ideal numbers by means of ideals. Because of the fact that Dedekind’s language is so much more familiar to modern readers we will use it rather than Kummer’s. The interested reader can consult Edward’s book [Ed1] for a discussion of Kummer’s point of view.

For the rings under consideration, Kummer proved that every ideal is the product of prime ideals in a unique way. He also defined the usual equivalence relation on ideals, proved the equivalence classes form a group, and that this group is finite. Let’s call the ideal class group of Z[ζ_{p}], Cl_{p}, and its order, h_{p}, the class number. We also refer to h_{p} as the class number of the field Q(ζ_{p}). Unique factorization holds for elements if and only if h_{p} = 1.

Another complication arises when p ≥ 5. Namely, the unit group will contain elements of infinite order. As is well known the units in Z[i] are the fourth roots of unity and those in Z[ω] are the sixth roots of unity. For p ≥ 5 we define the elements (set ζ_{p} = ζ)

ξ_{k} = \sqrt{\frac{ζ^{k} − 1}{ζ − 1} \frac{ζ^{−k} − 1}{ζ^{−1} − 1}} = \begin{matrix}sin & \frac{kπ}{p} \\ sin & \frac{π}{p} \end{matrix} k = 2, 3, …, \frac{p − 1}{2} .  (2)

These elements are easily seen to be units, called cyclotomic units, in Z[ζ_{p}]. Kummer shows that they are independent, i.e. they generate a free abelian group of rank \frac{p−3}{2}. Let C_{p} be the subgroup of the unit group E_{p} generated by the ξ_{k} and the roots of unity. C_{p} is called the group of cyclotomic units. Kummer shows that C_{p} is of finite index in E_{p} and gives the following beautiful interpretation of the index.

Theorem 2.1. The index [E_{p} : C_{p}] = h_{p}^{+} where h_{p}^{+} is the class number of Q(ζ_{p} + ζ_{p}^{−1}) = Q(ζ_{p})^{+} the maximal real subfield of Q(ζ_{p}).

Further, it turns out that h_{p}^{+} divides h_{p} so that we can define the integer h_{p}^{−} = h_{p}/h_{p}^{+}, called the relative class number. In the older literature h_{p}^{−} and h_{p}^{+} are referred to as the first and second factor of the class number respectively. Kummer also gives a beautiful and important formula for h_{p}^{−} which we shall now proceed to explain.

Let χ be a Dirichlet character modulo p. We say that χ is odd if χ(−1) = −1 and even if χ(−1) = 1. Define B_{1,χ} = p^{−1} ∑_{a=1}^{p−1} χ(a)a. It is easy to see that B_{1,χ} = 0 if χ is even and not trivial. On the other hand;

Theorem 2.2.

h_{p}^{−} = 2p ∏_{χ odd} (−\frac{1}{2}) B_{1,χ}.

The numbers B_{1,χ} are sometimes called generalized Bernoulli numbers. We will relate them to ordinary Bernoulli numbers a little later. First, we mention another of Kummer’s important theorems about the class number.

Theorem 2.3. If p divides h_{p}^{+} then p divides h_{p}^{−} .

Definition. A prime number p is called regular if p does not divide h_{p}, otherwise it is called irregular.

As was mentioned in the introduction, Kummer’s greatest contribution to FLT was to show that it is true for regular primes. We will sketch a proof of this in Section 3. For the remainder of this section we discuss Kummer’s criterion for regularity which allows one to actually compute whether a given prime is regular or not. We begin by recalling the definition of the Bernoulli numbers and some of their properties (see [BS], [IR], [Ri], or [W]).

The simplest way to define the Bernoulli numbers B_{n} is by way of the power series expansion

\frac{t}{e^{t} − 1} = ∑_{n=0}^{∞} B_{n} \frac{t^{n}}{n!} .

From this it is easy to derive the formula

(m + 1)B_{m} = − ∑_{k=0}^{m−1} \binom{m + 1}{k} B_{k}.

From this the Bernoulli numbers can be computed recursively. One finds B_{1} = −\frac{1}{2}, B_{2} = \frac{1}{6}, B_{3} = 0, B_{4} = −\frac{1}{30}, B_{5} = 0, B_{6} = \frac{1}{42}, etc. For n > 1 and odd one has B_{n} = 0. The even numbered Bernoulli numbers grow quite rapidly. In fact, |B_{2m}| > 2(m/πe)^{2m}.

The Bernoulli numbers are rational numbers whose denominators are known thanks to the theorem of von Staudt and Clausen. This asserts that the denominator of B_{n} is the product of those primes p such that p − 1 divides n. In particular, if p > 3 the numbers B_{2}, B_{4}, B_{6}, …, B_{p−3} are p-integral.

Let Z_{p} denote the p-adic numbers. It is well known that the unit group Z_{p}^{∗} contains the p−1’st roots of unity, and that there is a character ω : (Z/pZ)^{∗} → Z_{p}^{∗} with the property that ω(a) ≡ a (mod p) for all rational integers a prime to p. ω is an odd character of order p − 1. Using Theorem 2.2, one easily derives the following p-adic version.

Theorem 2.2 P.

h_{p}^{−} = 2p ∏_{{i=1 \atop i odd}}^{p−2} (−\frac{1}{2}B_{1,ω^{i}})

.

The equality here takes place inside Z_{p}.

To get Kummer’s criterion from this we need the following congruence which is proved in [W, Cor.5.15] using p-adic L-functions and in [Lg1, Theorem 2.5] using the theory of p-adic distributions. Because of its importance we give another, more elementary proof, in the appendix to this paper.

Proposition 2.3. If n is odd, and p − 1 does not divide n + 1, we have

B_{1,ω^{n}} ≡ \frac{B_{n+1}}{n + 1} (mod p).

We are now in a position to demonstrate the following wonderful result of Kummer.

Theorem 2.4. A prime number p is regular if and only if it does not divide the numerator of any of the Bernoulli numbers B_{2}, B_{4}, …, B_{p−3}.

Proof. By Theorem 2.3 it follows that p divides h_{p} if and only if p divides h_{p}^{−}. We will use the expression for h_{p}^{−} given by Theorem 2.2 P.

First consider the case i = p − 2. We have,

pB_{1,ω^{p−2}} = ∑_{a=1}^{p−1} ω^{p−2}(a)a ≡ ∑_{a=1}^{p−1} a^{p−1} ≡ p − 1 ≡ −1 (mod p) .

Thus,

h_{p}^{−} ≡ ∏_{{i=1 \atop i odd}}^{p−4} (−\frac{1}{2}B_{1,ω^{i}}) ≡ ∏_{{i=1 \atop i odd}}^{p−4} (−\frac{1}{2} \frac{B_{i+1}}{i + 1} ) (mod p) .

The result follows directly from this congruence.

By using this theorem it is possible to check a given prime for regularity. Kummer showed that the only irregular primes less than 100 are 37, 59, and 67. He later checked all the primes up to 164 and found that 101, 103, 131, 149, and 157 are the only additional irregular primes. At one point he thought that there are infinitely many regular primes. In 1915, K.L. Jensen proved that there are infinitely many irregular primes [Ri; Lecture VI, Section 4]. However, to this day it is not known if there are infinitely many regular primes. It is possible to give a probabilistic argument to show that over 60% of the primes should be regular.

Let’s assume that the probability that an even indexed Bernoulli number B_{2m} be divisible by p is \frac{1}{p}. If this is so, the probability that none of the numbers B_{2}, B_{4}, …, B_{p−3} be divisible by p is

(1 − \frac{1}{p})^{\frac{p−3}{2}} ≈ e^{− \frac{1}{2}} ≈ .6065 .

This estimate for the percentage of regular primes agrees very well with the experimental evidence [BCEM]. It would be nice to have a rigorous proof.

We end this section with two more of Kummer’s results. These two concern units. The first is fairly easy, but very useful. The second is quite deep, and is crucial to Kummer’s proof of FLT for regular primes.

Recall some simple facts about Q(ζ_{p}). The prime p is totally ramified in this field and the prime lying above (p) in Z[ζ_{p}] is (λ) where λ = ζ_{p} − 1. Note that ζ_{p} ≡ 1 (mod λ) and Z[ζ_{p}]/(λ) ≅ Z/pZ.

Proposition 2.5. Let u be a unit in Z[ζ_{p}]. Then u can be written as ζ_{p}^{i}e, where i is determined modulo p, and e is real.

Proof. Let bar denote, as usual, complex conjugation. Then, u/ū is a unit such that it and all its conjugates have absolute value 1. By a well known result u/ū is a root of unity. By ramification theoretic considerations the only roots of unity in Z[ζ_{p}] are ±ζ_{p}^{i} for 0 ≤ i ≤ p − 1. If u/ū = ζ_{p}^{i} find an integer j such that 2j ≡ i (mod p). Then one easily checks ζ_{p}^{−j}u is equal to its own conjugate, i.e. is real. The result follows in this case.

Suppose u/ū = −ζ^{i}. We will show this leads to a contradiction. Choosing j as above, and setting w = ζ_{p}^{−j}u, we find that w is a unit such that w̄ = −w. Thus w^{2} = −ww̄ = −v. It follows that −v ∈ Z[ζ_{p}]^{+} is a negative unit and so w generates the extension Q(ζ_{p})/Q(ζ_{p})^{+}. It would follow that this extension is unramified at p (recall p ≠ 2). However, it is ramified at p, so we have reached a contradiction.

Corollary. Let E_{p}^{+} be the real units in E_{p} and C_{p}^{+} be the subgroup of C_{p} generated by the cyclotomic units ±ξ_{k} (see equation (2) above). Then, E_{p}^{+}/C_{p}^{+} ≅ E_{p}/C_{p} and both groups have order h_{p}^{+}.

Proof. This follows directly from Theorem 2.1 and Proposition 2.5.

We remark that there is another way to finish the proof of the proposition. Suppose w is a unit such that w = −w̄. By the remarks preceeding the proposition, there is a rational integer M such that w ≡ M (mod λ). It follows that M ≡ −M (mod λ) and so 2M ≡ 0 (mod λ). Thus, λ divides M and so also w. However, w is a unit so this is a contradiction.

The final result we need is known as Kummer’s lemma. It is simple to state. Any unit in Q(ζ_{p}) which is congruent to a rational integer modulo p is a p’th power. The usual proof is by means of a close analysis of the unit group and especially of the local unit group in the completion of Q(ζ_{p}) at the prime (λ). This involves the p-adic logarithm, the local expansion of the p-adic logarithm of the cyclotomic units, etc. The analysis is nicely carried out in Section 6 of Chapter 5 in [B-S]. We give an alternate approach, suggested by Hilbert in his Zahlbericht [H], which depends on the study of ramification in Kummer extensions. This approach brings out more clearly the underlying role played by class field theory.

Lemma 2.6. Let β ∈ Z[ζ_{p}]. Suppose λ does not divide β and that x^{p} ≡ β (mod λ^{p}) is solvable. Let K be the field obtained by adjoining A = ^{p} \sqrt{β} to Q(ζ_{p}). Then the extension K/Q(ζ_{p}) is unramified at p.

Proof. Suppose α^{p} ≡ β (mod λ^{p}) and set τ = \frac{A−α}{λ}. Then τ is a root of the monic polynomial

f(x) = \frac{(λx + α)^{p} − β}{λ^{p}} .

Using the fact that p = uλ^{p−1} where u is a unit, we see that all the coefficients of f(x) are algebraic integers. Thus, τ is an algebraic integer. A short computation shows that all the coefficients of f′(x) are in (λ) except the constant term uα^{p} which is prime to (λ). Thus, f′(τ) is prime to (λ) and it follows that the relative discriminant is prime to (λ). The proof is complete.

Theorem 2.7. Let p be a regular prime and e a unit in Q(ζ_{p}) which is congruent to a p’th power modulo λ^{p}. Then, e is the p’th power of a unit.

Proof. Consider the extension L = Q(ζ_{p}, ^{p} \sqrt{e}) of Q(ζ_{p}). This extension is cyclic of degree 1 or p. Suppose the degree is p. By Lemma 2.6, the extension is unramified at p. Since e is a unit it is easy to see it is unramified at every other prime as well. Thus, it is an unramified, abelian extension of degree p and it follows by class field theory that p|h_{p}, which contradicts the assumption that p is regular. Thus, ^{p} \sqrt{e} ∈ Q(ζ_{p}) and the theorem follows.

The phrase “it follows by class field theory” can be avoided. The result needed follows from Theorem 94 of Hilbert’s Zahlbericht [H; page 155]. We sketch a short cohomological proof of this in the Appendix.

The usual form of Kummer’s Lemma can be deduced from Theorem 2.7 as follows. If e ≡ a (mod p) with a ∈ Z we have e = a + pα with α ∈ Z_{p}. Now, α ≡ b (mod λ) with b ∈ Z. Thus, e ≡ a + bp (mod λ^{p}). Let σ be an element of Gal(Q(ζ_{p})/Q). Then, e^{σ}/e ≡ 1 (mod λ^{p}) and is thus a p’th power by Theorem 2.7: e^{σ} = eη(σ)^{p}. Taking the product over all σ and remembering that the norm of a unit is 1, we find 1 = e^{p−1}η^{p} for a suitable η and finally, e = (eη)^{p}.

Section 3. FLT for Regular Primes and Certain Other Cases.

Having assembled a number of the powerful tools forged by Kummer, we will now give part of his proof that FLT is true for regular primes. Here is the statement of the full theorem.

Theorem 3.1. Let p be a regular prime. Then, the equation x^{p} + y^{p} = z^{p} has no solution with x, y, z ∈ Z[ζ_{p}] and xyz ≠ 0.

The proof is usually broken up into two parts. The first case is when λ does not divide xyz and the second case is when λ does divide xyz (recall, λ = ζ_{p} − 1). The first case is easier. If one confines one’s attention to Z rather than Z[ζ_{p}] proofs of the first case can be found in [B-S,I-R,W] and many other places. For a proof of the first case in Z[ζ_{p}] see [La] or [H]. We will concentrate on the second case.

It is interesting that Kummer made a simple error at the beginning of his proof of the second case. He asserts that it is no loss of generality to assume that x, y, z are pairwise relatively prime. This is certainly true over Z, but is false over Z[ζ_{p}] because there may be a common divisor which is not principal. Once this is realized it is not hard to alter Kummer’s proof so that it holds in full generality. Hilbert does so in Section 172 of [H]. See [La] for another presentation. We give Kummer’s proof of the more restricted result because it is relatively short and the main ideas show through more clearly.

Theorem 3.1’. Let p be a regular prime. Then, the equation

x^{p} + y^{p} = z^{p}

has no solution with x, y, z ∈ Z[ζ_{p}], λ|xyz, and x, y, z pairwise relatively prime.

Proof. Assume x, y, z is such a solution. It is no loss of generality to assume that λ|z. It follows that λ does not divide x or y. Write z = λ^{m}z_{0} with (λ, z_{0}) = 1. Then, x, y, z_{0} is a solution to X^{p} + Y^{p} = λ^{mp}Z^{p} with (xyz_{0}, λ) = 1, x, y, z pairwise relatively prime, and m ≥ 1. Let u be a unit in Z[ζ_{p}] . We will show that there are no solutions x, y, z to

X^{p} + Y^{p} = uλ^{mp}Z^{p}  (∗)

with x, y, z pairwise relatively prime, (xyz, λ) = 1, and m ≥ 1. This will prove the theorem.

The strategy is this. Assume such a solution exists. One shows that, in fact, m must be greater than 1. Then one finds a solution of the same type to a similar equation but with m replaced by m − 1. This is yields a contradiction via “infinite descent”.

We need a Lemma.

Lemma 3.2. Let v ∈ Z[ζ_{p}] with (v, λ) = 1. Then, there is a rational integer k such that ζ_{p}^{k}v ≡ a (mod λ^{2}) with a ∈ Z.

Proof. Since the residue class field modulo λ has p elements, we may write v ≡ m + nλ (mod λ^{2}) where m, n ∈ Z and (m, p) = 1. Now, ζ_{p}^{k} = (1 + λ)^{k} ≡ 1 + kλ (mod λ^{2}). Thus, ζ_{p}^{k}v ≡ m + (n + km)λ (mod λ^{2}). Choose k to be a solution of n + mx ≡ 0 (mod p) and the Lemma follows.

Now, assume x, y, z is a solution of equation (∗) above with (xyz, λ) = 1 and x, y, z pairwise relatively prime. By Lemma 3.2 we can assume that x, y ≡ a, b (mod λ^{2}) where a, b ∈ Z. We have

∏_{i=0}^{p−1}(x + ζ_{p}^{i}y) = uλ^{mp}z^{p}.  (∗∗)

It follows that λ must divide at least one term on the left hand side of this equation, and consequently it must divide all the terms. Since x + y ≡ a + b (mod λ^{2}), we must have p | a + b and so λ^{2} | x + y. From equation (∗∗) it now follows that λ^{p+1} divides the left hand side, and so m > 1, which is our first goal.

We have x + ζ_{p}^{i}y = x + y + (ζ_{p}^{i} − 1)y and so for i > 0 x + ζ_{p}^{i}y is exactly divisible by λ. Thus, passing to ideals, we find (x + y) = (λ)^{p(m−1)+1}C_{0} and (x + ζ_{p}^{i}y) = (λ)C_{i} for i = 1, 2, …, p−1, where the C_{i} for i = 0, 1, …, p−1 are pairwise relatively prime principal ideals each of which is prime to (λ). From equation (∗∗) we deduce C_{0}C_{1} … C_{p−1} = (z)^{p}. It follows that each C_{i} is a p’th power, i.e C_{i} = D_{i}^{p}. Since D_{i}^{p} is a principal ideal and p is a regular prime, we deduce that D_{i} is a principal ideal, i.e. D_{i} = (w_{i}) for w_{i} ∈ Z[ζ_{p}]. Returning to the level of elements, we see that there are units u_{0}, u_{1}, u_{2} such that

x + y = u_{0}λ^{p(m−1)+1}w_{0}^{p}, x + λy = u_{1}λw_{1}^{p}, x + λ^{2}y = u_{2}λw_{2}^{p}.

Eliminating x and y from these three equations and dividing the result by u_{1} we find units e_{2} and e such that

w_{1}^{p} + e_{2}w_{2}^{p} = eλ^{(m−1)p}w_{0}^{p}.

Taking congruences modulo λ^{p} we see that e_{2} is congruent to a p’th power modulo λ^{p}. By Theorem 2.7, e = f^{p} for some f ∈ Z[ζ_{p}]. Set x′ = w_{1}, y′ = fw_{2}, and z′ = w_{0} we see that x′, y′, z′ is a solution to X^{p} + Y^{p} = eλ^{(m−1)p}Z^{p} for which (x′y′z′, λ) = 1 and x′, y′, z′ are pairwise relatively prime. We have reached our second goal and so completed the proof of Theorem 3.1’.

Kummer went well beyond the case of regular primes in his attempt to prove FLT in general. He produced some explicit cyclotomic units E_{i} for i = 2, 4, …, p − 3 and stated that FLT is true if the following conditions hold; h_{p}^{−} is divisible by p but not p^{2}, and for i = 2, 4, …, p − 3, B_{pi} is not divisible by p^{3} and E_{i} is not a p’th power. He then verified on the basis of this result and Theorem 3.1 that FLT holds for all primes less than 100.

Kummer’s work was reconsidered by H.S. Vandiver in the 1920’s. He found some problems with the proof of the above mentioned result which he was able to fix. He went on to improve upon Kummer’s work in several ways. For example, he proved the following theorem [V], [W; Theorem 9.4].

Theorem 3.3. Suppose B_{pi} is not divisible by p^{3} for i = 2, 4, …, p − 3 and that h_{p}^{+} is not divisible by p. Then, the second case of FLT is true for p.

In Chapter 9 of [W] there are further results of Vandiver which give rational criteria for proving the second case of FLT. See, in particular, Theorem 9.5. The first case of FLT can also be tested by even simpler rational criteria (see the article by G. Cornell in this volume). Thus, it became possible to test FLT computationally. Vandiver and his students, using desk calculators, extended Kummer’s verification to all primes less than 620. In 1954, using a computer, Vandiver, D.H. Lehmer, and E. Lehmer verified FLT for all primes up to 2, 000. In 1955, J.L. Selfridge, C.A. Nicol, and H.S. Vandiver verified FLT up to 4, 001. In 1976, S.Wagstaff showed FLT is true for all p < 125, 000. In 1993 it was shown that FLT is true for all primes up to 4 million [BCEM]. This is the largest bound achieved before FLT was proven to hold for all primes p and so all n > 2.

Vandiver (1882-1973) made many contributions to FLT and other parts of number theory. For references to some of his work on FLT see his interesting expository article (and the short follow up article) [V]. According to an interesting obituary notice written by D.H. Lehmer [Lmr], Vandiver never graduated from high school. He spent most of his professional life at the University of Texas. He is the only American mentioned in E. Landau’s monumental three volume treatise on number theory, Vorlesungen über Zahlentheorie.

Vandiver conjectured that h_{p}^{+} is not divisible by p for all primes p. This conjecture is referred to simply as Vandiver’s conjecture. It has pointed out that Kummer made the conjecture many years earlier in a letter to Kronecker where he refers to it as a “noch zu beweisenden Satz” (Kummer’s Collected Works, vol. 1, page 85). In any case, it has held up well. In [BCEM] it is verified for all primes less than 4 million. Larry Washington has produced a probabilistic argument [W; page 159] which shows that the number of exceptions to Vandiver’s conjecture up to a given bound x could be approximately \frac{1}{2} log log x. For x = 4, 000, 000 this is approximately 1.361 so the fact that no counter-example has shown up is perhaps not surprising. On the other hand Washington’s reasoning rests on certain randomness assumptions which may or may not hold. In any case, the conjecture is true very often! This is good because, as we shall see in the next section, Vandiver’s conjecture has very interesting implications.

Section 4. The Structure of the p-Class Group .

A prime is irregular if A_{p}, the p-part of the class group of Q(ζ_{p}), is non-trivial. Much work has been devoted to understanding the structure of A_{p} beginning with Kummer. Important contributions have been made by people like Hilbert, Herbrand, Leopoldt, Iwasawa, Ribet, Mazur, and Wiles, among others. In this section we will review some of this work. Among other things we will show that if one accepts the Vandiver conjecture is true then it is possible to give a completely satisfying description of the structure of A_{p}.

We begin with a few preliminary remarks. It is convenient to write the group operation in A_{p} additively. Also, since p is fixed in this discussion, we will write A instead of A_{p}. Since A is a torsion p-group, we may consider it as a Z_{p} module. It is also a module for the Galois group G_{p} = Gal(Q(ζ_{p})/Q). Thus, A is a module over the group ring Z_{p}[G_{p}]. As is well known, (Z/pZ)^{∗} is isomorphic to G_{p} where the automorphism corresponding to a, σ_{a}, takes ζ_{p} to ζ_{p}^{a}. Let ω be the p-adic valued character introduced in Section 2, and for each i such that 0 ≤ i < p − 1 define

ϵ_{i} = \frac{1}{p − 1} ∑_{a} ω(a)^{−i}σ_{a} ∈ Z_{p}[G_{p}].

Here as elsewhere in this section the summation goes from a = 0 to a = p−2. It is easy to check that these elements constitute a complete set of mutually orthogonal idempotents in the group ring. Define A_{i} = ϵ_{i}A. Then,

A = ⨁_{i} A_{i} and if x ∈ A_{i}, σ_{a}x = ω(a)^{i}x.

Because of this decomposition, to understand the structure of A it suffices to understand the structure of each A_{i}.

The automorphism σ_{−1} is simply complex conjugation. Since ω(−1) = −1 it follows that complex conjugation fixes A_{i} if i is even, and acts as multiplication by −1 if i is odd. It is not hard to show that the part of A which is fixed by complex conjugation is isomorphic to the p-part of the class group of Q(ζ_{p})^{+}. Thus, if Vandiver’s conjecture is true, A_{i} = (0) for i even.

A final preliminary comment is that ϵ_{0} is a constant times the norm map, and the norm map annihilates the class group. It follows that ϵ_{0} simultaneously annihilates and fixes A_{0}. The conclusion is that A_{0} = (0).

Do any other elements of the group ring Z[G_{p}] annihilate A? This question is the key to the deeper part of the theory. The answer is that yes, there are other elements beside the norm map which annihilate the class group. The honor of being the first to see this goes (once again) to Kummer. Let l be a rational prime such that l ≡ 1 (mod p), L a prime lying above it in Q(ζ_{p}), and let χ be an appropriately chosen character of (Z/lZ)^{∗} of order p. Form the Gauss sum

G(χ) = ∑_{x=0}^{l−1} χ(x)ζ_{l}^{x}.

Then, it is easy to show G(χ)^{p} ∈ Q(ζ_{p}). What Kummer did (his notation was different) was to determine the prime decomposition of this element. He showed,

(G(χ)^{p}) = L^{∑ aσ_{a}^{−1}}.

Kummer also showed that in every ideal class there are ideals all of whose prime factors have absolute degree 1. It follows from these considerations that ∑ aσ_{a}^{−1} annihilates the class group of Q(ζ_{p}). Somewhat later, 1890 to be precise, L. Stickelberger generalized all these considerations to the fields Q(ζ_{m}) with m arbitrary. Complete proofs of Stickelberger’s theorem can be found in [IR] and [W]. We will deal only with the case m = p. Let us state the result we will need.

Theorem 4.1. Define θ = \frac{1}{p} ∑ aσ_{a}^{−1}. Then, pθ annihilates the class group. Further, suppose (b, p) = 1. Then, (σ_{b} − b)θ ∈ Z_{p}[G_{p}] and (σ_{b} − b)θ annihilates the class group.

Corollary. A_{0} = (0) and A_{1} = (0). For i ≥ 2, B_{1,ω^{−i}} annihilates A_{i}. (Note that this is only interesting for i odd since the generalized Bernoulli numbers are zero for even characters).

Proof. We have already shown that A_{0} = (0). Now, pθ acts on A_{1} by multiplication by ∑ aω(a)^{−1} = ∑ aω(a)^{p−2} ≡ ∑ a^{p−1} ≡ p − 1 ≡ −1 (mod p). This shows that pθ acts on A_{1} by multiplication by a p-adic unit. On the other hand it annihilates A_{1} by the theorem. Thus, A_{1} = (0).

Assume now that i ≥ 2. Then, (σ_{b} − b)θ acts on A_{i} by multiplication by (ω(b)^{i} − b)B_{1,ω^{−i}}. Choose b to be a primitive root modulo p. Then, ω(b)^{i} − b ≡ b(b^{i−1} − 1) ≢ 0 (mod p). This factor is therefore a p-adic unit and so B_{1,ω^{−i}} annihilates A_{i} as asserted. (It is easy to check using the defining property of ω that B_{1,ω^{−i}} ∈ Z_{p} for i > 1).

We can now prove the following important theorem of J. Herbrand which appeared (posthumously) in 1932 [He].

Theorem 4.2. Let i be odd and 3 ≤ i ≤ p − 2. Set j = p − i. If p does not divide B_{j} then A_{i} = (0).

Proof. By the above corollary, B_{1,ω^{−i}} annihilates A_{i}. By Proposition 2.3,

B_{1,ω^{−i}} = B_{1,ω^{p−1−i}} ≡ \frac{B_{p−i}}{p − i} (mod p).

Thus, if p doesn’t divide B_{p−i} = B_{j}, it follows that B_{1,ω^{−i}} is a p-adic unit and so A_{i} = (0).

Herbrand also proved the converse to this theorem, but under the assumption that Vandiver’s conjecture is true. In 1976, Ribet proved the converse is true unconditionally [R].

Theorem 4.3. Let i be odd and 1 ≤ i ≤ p − 2. Set j = p − i. If p divides B_{j} then A_{i} ≠ 0.

Ribet’s methods are completely different from those used by Herbrand which were classical, descending in a direct line from Kummer. Ribet used the rapidly developing arithmetic theory of modular forms and Galois representation theory, the same tools used eventually to successfully attack FLT itself.

Herbrand’s theorem and its converse are somewhat qualitative in nature. It provides a rational criterion for determining when A_{i} is trivial or not. In 1980, Wiles proved the following, more quantitative, result [Wi].

Theorem 4.4. Let i be odd and 3 ≤ i ≤ p − 2. Assume that A_{i} is cyclic. Then #(A_{i}) = p^{m_{i}} where m_{i} = ord_{p}(B_{1,ω^{−i}}).

Once again, the methods used were the modern ones involving modular forms, modular curves, and representation theory. The result is very nice, but is conditional on the hypothesis of cyclicity. This was removed a few years later in 1984 when B. Mazur and A. Wiles published a proof of the main theorem of Iwasawa theory for abelian extensions of Q. See [MW]. As a corollary of their work, they deduce the following unconditional result.

Theorem 4.5. Let i be odd and 3 ≤ i ≤ p − 2. Then #(A_{i}) = p^{m_{i}} where m_{i} = ord_{p}(B_{1,ω^{−i}}).

This result makes no assertion as to whether A_{i} is cyclic or not. It gives its size but not its structure.

The Mazur-Wiles result once again uses sophisticated modern methods involving the theory of modular forms, modular curves, and modular Jacobians. A different approach was found by V.A. Kolyvagin who invented the method of Euler systems which is in principal much more elementary. The application of these methods to Iwasawa theory is exposited in an appendix by K. Rubin to the new edition of Lang’s book on cyclotomic fields [Lg1].

We will end this section by showing how “classical” methods together with the conjecture of Vandiver enables one to give a complete structure theorem for A. In essence, this is due to Herbrand. However, we will make good use of the “Spiegelungssatz” which is due to H.W. Leopoldt [Le] (1958). Special cases were proved independently by Iwasawa. We state such a special case next. The proof involves Kummer theory and class field theory. See Section 10.2 of [W]. If B is a finite abelian p-group, define rank B to be the dimension over Z/pZ of B/pB. This can be shown to be the minimal number of generators for B.

Theorem 4.6. Suppose j is even, 1 < i, j < p − 1, and i + j = p. Then,

rank A_{j} ≤ rank A_{i} ≤ rank A_{j} + 1.

Corollary. Assuming the Vandiver conjecture, A_{j} = (0) for j even and A_{i} is cyclic for i odd.

It should also be remarked that the first inequality of the Theorem can be viewed as a refinement of Kummer’s result, Theorem 2.3, that if p|h_{p}^{+} then p|h_{p}^{−}.

Define A^{+} = {x ∈ A| σ_{−1}x = x} and A^{−} = {x ∈ A| σ_{−1}x = −x}. Then A = A^{+} +A^{−} and A^{+} = (0) if Vandiver’s conjecture holds.

We are now in a position to give a satisfying structure theorem for A_{p} under the assumption that Vandiver’s conjecture is correct.

Theorem 4.7. Assume Vandiver’s conjecture is true. Then,

A = A^{−} = A_{3} ⊕ A_{5} ⊕ … ⊕ A_{p−2}.

For i odd, 3 ≤ i ≤ p − 2, set m_{i} = ord_{p}B_{1,ω^{−i}}. Then, A_{i} ≃ Z_{p}/B_{1,ω^{−i}}Z_{p} ≃ Z/p^{m_{i}}Z. Finally, m_{i} > 0 if and only if p|B_{p−i}.

Proof. Vandiver’s conjecture implies A^{+} = (0) and so A = A^{−}. By the Corollary to Theorem 4.1, A_{1} = (0), so the first assertion follows. By the same Corollary, B_{1,ω^{−i}} annihilates A_{i} so A_{i} is a Z_{p}/B_{1,ω^{−i}}Z_{p} ≃ Z/p^{m_{i}}Z module. By the Corollary to Theorem 4.6 we have that A_{i} is cyclic. It follows that #(A_{i}) ≤ p^{m_{i}}. Taking the product over all i, we find #(A^{−}) ≤ p^{∑ m_{i}}. We claim that this inequality is actually an equality. Consider the class number formula given in Theorem 2.2 P, and recall that pB_{1,ω^{−1}} is a p-adic unit. It follows that ord_{p}(h_{p}^{−}) = m_{3} + m_{5} + … + m_{p−2}. Since ord_{p}#(A^{−}) = ord_{p}(h_{p}^{−}) we have proven our claim. This shows that #(A_{i}) = p^{m_{i}} for all i. Since A_{i} is a cyclic Z/p^{m_{i}}Z module, the conclusion is that A_{i} ≃ Z/p^{m_{i}}Z.

The final assertion follows from the congruence B_{1,ω^{−i}} ≡ B_{p−i}/(p − i) (mod p).

The index of irregularity of a prime p is defined to be the number of Bernoulli numbers in the set {B_{2}, B_{4}, …, B_{p−3}} which are divisible by p. Call this number r_{p}. The above theorem shows that when Vandiver’s conjecture is true the rank of A_{p} is equal to r_{p}. By [BCEM] we know that Vandiver’s conjecture is true for all primes up to 4 million. They also are able to verify that in this range A_{p} is an elementary p-group. Thus,

A_{p} ≃ (Z/pZ)^{r_{p}}

for all primes less than 4 million. They also show a similar result for the p-part of the class group of Q(ζ_{p^{n}}).

It is interesting to ask about the properties of the invariant r_{p}. For example, can it be arbitrarily large? If one assumes that the probability that a given Bernoulli number is divisible by p is 1/p the probability (assuming some independencies) that r_{p} = k is given by

\binom{\frac{p−3}{2}}{k} (\frac{1}{p})^{k} (1 − \frac{1}{p})^{\frac{p−3}{2} −k} ≈ \frac{e^{− \frac{1}{2}}}{2^{k}k!}.

(The left hand side approaches the right hand side as p → ∞.) The calculations of [BCEM] show that this is in excellent agreement with the facts for k = 0, 1, 2, 3, 4, 5 for the set of prime less than 4 million. The largest value of r_{p} found was 7 which occurs only once for p = 3, 238, 481. If one accepts the validity of the probabilistic calculation just given it would follow that r_{p} can be arbitrarily large, but no proof of this is known. It is also unknown if the exponent of the group A_{p} can be arbitrarily large.

Section 5. Suggested Readings .

In this brief final section we will point to a few sources for further reading.

Since so much of our story has concerned Kummer it is perhaps most suitable to begin with the first volume of his collected works [Ku]. The first volume contains his contributions to number theory. The introductory essay by A. Weil is very enlightening both for its assessment of the significance of Kummer’s contributions and as a guide to the papers themselves. These volume were first published in 1975 and there are two informative reviews to recommend; one by Edwards [Ed4] and the other by B. Mazur [M]. Mazur’s review is especially valuable for connecting Kummer’s work to modern developments, much of which is due to Mazur himself.

As for FLT itself, there is the interesting, but mostly non-mathematical, book by E.T. Bell [B]. More substantial is the book by Edwards [Ed1]. In spite of its title, the emphasis is not so much FLT as the development of algebraic number theory in the hands of Kummer. In particular, it elaborates Kummer’s theory of ideal numbers in much greater detail than any other work available in English. Also by Edwards are the interesting papers [Ed2] and [Ed3], already mentioned earlier, which debunk the story that Kummer thought he had proven FLT but had mistakenly assumed unique factorization in Z[ζ_{p}]. The best source for covering the spectrum of work done on FLT and related topics up through 1978 is the book of Ribenboim [Ri]. It is erudite and readable at the same time. Moreover the bibliography is very extensive and useful.

The area of p-adic analysis is a modern development with origins in Kummer’s arithmetic work. A good introduction is the book of N. Koblitz [Ko]. More advanced, and more inclusive in its coverage is the book of Lang [Lg1]. As has been pointed out earlier, the new edition of this book contains a valuable appendix by Rubin which gives a proof of the main theorem of Iwasawa theory using the methods of Kolyvagin.

For the purposes of finding the proofs of most of the facts related in this paper, the best reference is the book by L. Washington [W]. A second edition of this book is in production and it will contain a number of interesting new results , e.g. a proof of the converse to Herbrand’s theorem which uses Kolyvagin’s Euler systems and avoids the more sophisticated methods used by Ribet. Of course, it should not be forgotten that these more sophisticated methods are what underlie the magnificent proof found by Wiles of FLT, a proof that eluded mathematicians for hundreds of years!

Appendix.

We will sketch the proof of two important results mentioned in the text. The first is a congruence which, in essence, is due to Kummer. The second is Hilbert’s theorem 94, which is not as famous as Hilbert’s theorem 90 (due to Kummer), but nevertheless of great importance in the history of class field theory. Our proof will use some properties of cohomology of groups.

Theorem A1. Let p be an odd prime, n > 0 an odd integer, and assume that p − 1 does not divide n + 1. Then,

B_{1,ω^{n}} ≡ \frac{B_{n+1}}{n + 1} (mod p).

Proof. It is well known and easily proved that ω(a) = lim_{k→∞} a^{p^{k}} where a is any p-adic integer and the limit is taken p-adically. It follows that for k sufficiently large

pB_{1,ω^{n}} ≡ ∑_{a=1}^{p−1} a^{1+np^{k}} ≡ pB_{1+np^{k}} (mod p^{2}).  (∗)

The last congruence follows from the Corollary to Proposition 15.2.2 of [IR] and the observation that 1 + np^{k} ≡ 1 + n (mod p − 1). The Kummer’s congruences [IR; Thm.5, Ch 15], imply

\frac{B_{1+np^{k}}}{1 + np^{k}} ≡ \frac{B_{n+1}}{n + 1} (mod p).  (∗∗)

Dividing congruence (∗) by p and using congruence (∗∗) yields the result.

The second result we want to prove is Hilbert’s Theorem 94. We actually prove a slightly more general result which, given the cohomological tools we shall use, is no harder to prove (For further results using this type of analysis see [CR]).

Theorem A2. Let L/K be a cyclic unramified extension of algebraic number fields of odd degree n. Then, the order of the kernel of the natural map from Cl_{K} to Cl_{L} is divisible by n. In particular, the class number of K is divisible by n. (Here, Cl_{K} and Cl_{L} denote the class groups of K and L respectively).

Proof. Let D_{L}, P_{L}, and U_{L} denote the divisors, principal divisors, and units of L respectively, with similar notation for the field K. Let G = Gal(L/K). Consider the two exact sequences,

(0) → U_{L} → L^{∗} → P_{L} → (0)  (1)
(0) → P_{L} → D_{L} → Cl_{L} → (0)  (2)

Using Hilbert’s Theorem 90 and equation (1), we find H^{1}(G, U_{L}) = P_{L}^{G}/im(P_{K}). From equation (2) we derive

(0) → P_{L}^{G}/im(P_{K}) → D_{L}^{G}/im(P_{K}) → CL_{L}^{G}.

Since L/K is unramified, D_{L}^{G} = im(D_{K}). Combining these remarks we derive the following exact sequence,

(0) → H^{1}(G, U_{L}) → Cl_{K} → Cl_{L}^{G} .  (3)

.

For any G module M, the Herbrand quotient of M is defined to be

h(M) = \frac{#H^{0}(G, M)}{#H^{1}(G, M)} .

Given the hypotheses of the theorem it is well known that h(U_{L}) = n^{−1}. For the proof, which depends on an analysis of the structure of U_{L} ⊗ Q as a Q[G] module, see [Lg2], [AT], or the article of J. Tate in [CF]. It follows immediately that n divides the order of H^{1}(G, U_{L}) which by equation (3) is isomorphic to the kernel of the natural map from Cl_{K} to Cl_{L}. This completes the proof.

If n is even, it is still possible to derive results of a similar nature by making more restrictive hypotheses. For example, as Hilbert remarks, if n = 2 the result remains valid if we assume that every real prime of K splits in L.

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