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Algebraic Numbers
By Barry Mazur
The roots of our subject go back to ancient Greece while its branches touch almost all aspects of contemporary mathematics. In 1801 the Disquisi- tiones Arithmeticae of Carl Friedrich Gauss was first published, a “founding treatise,” if ever there was one, for the modern attitude towards number theory. Many of the still unachieved aims of cur- rent research can be seen, at least in embryonic form, as arising from Gauss’s work. This article is meant to serve as a companion to the reader who might be interested in learning, and thinking about, some of the classical theory of alge- braic numbers. Much can be understood, and much of the beauty of algebraic numbers can be appreci- ated, with a minimum of theoretical background. I recommend that readers who wish to begin this journey carry in their backpacks Gauss’s Disqui- sitiones Arithmeticae as well as Davenport’s The Higher Arithmetic (1992) which is one of the gems of exposition of the subject, and which explains the founding ideas clearly and in depth using hardly anything more than high-school mathematics.
1 The Square Root of 2
The study of algebraic numbers and algebraic integers begins with, and constantly reverts back to, the study of ordinary rational numbers and ordinary integers. The first algebraic irrationali- ties occurred not so much as numbers but rather as obstructions to simple answers to questions in geometry. That the ratio of the diagonal of a square to the length of its side cannot be expressed as a ratio of whole numbers is purported to be one of the vexing discoveries of the early Pythagoreans. But this very ratio, when squared, is 2:1. So we might— and later mathematicians certainly did—deal with it algebraically. We can think of this ratio as a cipher, about which we know nothing beyond the fact that its square is 2 (a viewpoint taken toward algebraic number by Kronecker, as we shall see below). We can write
2 in various forms, e.g. √ 2 = | 1 − i|, (1.1)
and we can think of 1−i = 1−e^2 πi/^4 as the world’s simplest trigonometric sum; we shall see general- izations of this for all quadratic surds below. We can also view
2 as a limit of various infinite sequences, one of which is given by the elegant continued fraction √ 2 = 1 +
2 + 2+^1..
Directly connected to this continued fraction (1.2) is the Diophantine equation
2 X^2 − Y 2 = ± 1 (1.3)
known as the Pell equation. There are infinitely many pairs of integers (x, y) satisfying this equa- tion, and the corresponding fractions y/x are pre- cisely what you get by truncating the expression in (1.2). For example, the first few solutions are (1, 1), (2, 3), (5, 7), and (12, 17), and 3 2 = 1 +^
1 2 = 1.^5 , 7 5 = 1 +^
17 12 = 1 +^
2 + 2+^1
2
Replace the ±1 on the right-hand side of (1.3) by zero and you get 2X^2 − Y 2 = 0, an equation all of whose positive real-number solutions (X, Y ) have the ratio Y /X =
2, so it is easy to see that the sequence of fractions (1.4) (these being alternately larger and smaller than
2 = 1. 414... ) converges to
2 in the limit. Even more striking is that (1.4) is the full list of fractions that best approx- imate
- (A rational number a/d is said to be a best approximant to a real number α if a/d is closer to α than any rational number of denomina- tor smaller than or equal to d.) To deepen the pic- ture, consider another important infinite expres- sion, the conditionally convergent series
log(
= 1 − 13 − 15 + 17 + 19 + · · · ± (^1) n + · · · , (1.5) Here the n range over positive odd numbers, and the sign of the term ± (^) n^1 is plus if n has a remain- der of 1 or 7 when divided by 8, and it is minus otherwise, i.e. if n has a remainder of 3 or 5 when divided by 8. This elegant formula (1.5), which you
Figure 1.1. The outer rectangle has its height-to-width ratio equal to the golden mean. If you remove a square from it as indicated in the figure, you are left with a rectangle that has the golden mean as its width-to- height ratio. This procedure is of course repeatable.
are invited to “check out” at least to one digit accuracy with a calculator, is an instance of the powerful and general theory of analytic formulas for special values of L-functions, which plays the role of a bridge between the more algebraic and the more analytic sides of the story. When we allude to this, below, we will call it “the analytic formula,” for short.
2 The Golden Mean
If you are looking for quadratic irrationalities that have been the subject of geometric fascination through the ages, then
2 has a strong competi- tor in the number 12 (1 +
5), known as the golden mean. The ratio 12 (1 +
5):1 gives the proportions of a rectangle with the property that when you remove a square from it, as in Figure 1.1, you are left with a smaller rectangle whose sides are in the same proportion. Its corresponding trigonometric sum description is
1 2 (1 +^
- = 12 + cos 25 π − cos 45 π. (2.1)
Its continued-fraction expansion is
1 2 (1 +^
1 + 1+^1.
where the sequence of fractions obtained by suc- cessive truncations of this continued fraction, y x =^
1 1 ,^
2 1 ,^
3 2 ,^
5 3 ,^
8 5 ,^
13 8 ,^
21 13 ,^
34 21 ,... ,^ (2.3) is the sequence of best rational-number approxi- mants to 1 2 (1 +^
where “best” has the sense already mentioned. For example, the fraction 34 21
1+ (^) 1+ 11 1+ (^) 1+^11 1 equals 1. 619047619047619047... and is closer to the golden mean than any fraction with denomi- nator less than 21. Nevertheless, the exclusive appearance of 1s in this continued fraction^1 can be used to show that, among all irrational real numbers, the golden mean is the number that is, in a specific technical sense, least well approximated by rational numbers. Readers familiar with the sequence of Fibonacci numbers will recognize them in the successive denominators of (2.3)—and in the numerators as well. The analogue to equation (1.2) is X^2 + XY − Y 2 = ± 1. (2.4)
This time, if you replace the ±1 on the right-hand side of the equation by 0, you get the equation X^2 + XY − Y 2 = 0, whose positive real-number solutions (X, Y ) have the ratio Y /X = 12 (1+
that is, the golden mean. And now the numerators and denominators y, x that appear in (2.3) run through the positive integral solutions of (2.4). The analogue of formula (1.4) (the “analytic formula”) for the golden mean is the conditionally convergent infinite sum 2 log( 12 (1 +
= 1− 12 − 13 + 14 + 16 +· · ·± (^1) n +· · · , (2.5) (^1) The continued-fraction expansion of any real quadratic algebraic number has an eventually recurring pattern in its entries, as is vividly exhibited by the two examples (1.2) and (2.2) given above.
(with ordinary integers as coefficients) of 1 and one of these fundamental quadratic algebraic integers.
4 Rings and Fields
I think that one of the big early advances in math- ematics is the now-current, universal recognition of the importance of studying the properties of col- lections of mathematical objects, and not just the objects in isolation. A ring R of complex numbers is a collection of them that contains 1 and is closed under the operations of addition, subtraction, and multiplication. That is, if a, b are any two num- bers in R, a ± b and ab must also be in R. If such a ring R has the further property that it is closed under division by nonzero elements (i.e. if a/b is again in R whenever a and b are, and b = 0), then we say that R is a field. (These concepts are dis- cussed further in fields and rings and ideals.) The ring Z of ordinary integers, { 0 , ± 1 , ± 2 ,... } is our “founding example” of a ring; visibly, it is the smallest ring of complex numbers. The collection of all real or complex numbers that are integral linear combinations of 1 and τd is closed under addition, subtraction, and multiplica- tion, and so is a ring, which we denote by Rd. That is, Rd is the set of all numbers of the form a + bτd where a and b are ordinary integers. These rings Rd—our first, basic, examples of rings of algebraic integers beyond that prototype, Z—are the most important rings that are receptacles for quadratic irrationalities. Every quadratic irrational algebraic integer is contained in exactly one Rd. For example, when d = −1 the corresponding ring R− 1 , usually referred to as the ring of Gauss- ian integers, consists of the set of complex numbers whose real and imaginary parts are ordinary inte- gers. These complex numbers may be visualized as the vertices of the infinite tiling of the complex plane by squares whose sides have length 1 (see Figure 1.2). When d = −3 the complex numbers in the cor- responding ring R− 3 may be visualized as the ver- tices of the regular hexagonal tiling of the complex plane (see Figure 1.3). With the rings Rd in hand, we may ask ring- theoretic questions about them, and here is some of the standard vocabulary useful for this. A unit u in a given ring R of complex numbers is a number in R whose reciprocal 1/u is also in R; a prime
−2 + i −1 + i + i 1 + i 2 + i
−2 − i −1 − i − i 1 − i 2 − i
−2 −1 0 1 2
Figure 1.2. The Gaussian integers are the vertices of this lattice of squares tiling the complex plane.
(or synonymously, an irreducible) element in R is a nonunit that cannot be written as the product of two nonunits in R. A ring of complex numbers R has the unique factorization property if every nonzero, nonunit, algebraic number in R can be expressed as a product of prime elements in exactly one way (where two factorizations are counted as the same if one can be obtained from the other by rearranging the order in which the primes appear and multiplying them by units). In the prototype ring Z of ordinary integers, the only units are ±1. The fundamental fact that any ordinary integer greater than 1 can be uniquely expressed as a product of (positive) prime num- bers (that is, that Z enjoys the unique factorization property) is crucial for much of the number theory done with ordinary integers. That this unique fac- torization property for integers actually required proof was itself a hard-won realization of Gauss, who also provided its proof (see fundamental theorem of arithmetic). It is easy to see that there are only four units in the ring R− 1 of Gaussian integers, namely ±1 and ±i; multiplication by any of these units effects a symmetry of the infinite square tiling (Figure 1. above). There are only six units in the ring R− 3 , namely ±1, ± 12 (1 +
−3) and ± 12 (1 −
−3); mul- tiplication by any of these units results in a sym- metry of the infinite hexagonal tiling (Figure 1. above).
−1 + τ (^) − 3 τ− 3
−1 0 +
− τ− 3 1 − τ− 3
Figure 1.3. The elements of the ring R− 3 are the ver- tices of this lattice of hexagons tiling the complex plane.
Fundamental to understanding the arithmetic of Rd is the following question: which ordinary prime numbers p remain prime in Rd and which ones fac- torize into products of primes in Rd? We will see shortly that if a prime number does factorize in Rd, it must be expressible as the product of pre- cisely two prime factors. For example, in the ring of Gaussian integers, R− 1 , we have the factorizations
2 = (1 + i)(1 − i), 5 = (1 + 2i)(1 − 2i), 13 = (2 + 3i)(2 − 3i), 17 = (1 + 4i)(1 − 4i), 29 = (2 + 5i)(2 − 5i), .. .
where all the Gaussian integer factors in brackets above are prime in the ring of Gaussian integers. Let us say that an odd prime p splits in R− 1 if it factorizes into a product of at least two primes and remains prime if it does not do so. As we shall soon see, the officially agreed-upon definitions of split- ting and remaining prime for more general rings of algebraic integers (even ones of the form Rd) are worded slightly, but very significantly, differently from the way we have just defined these concepts in the ring R− 1 of Gaussian integers. (Note that we have excluded the prime p = 2 from the above dichotomy. This is because 2 ramifies in R− 1 ; for a discussion of this concept see Section 7 below.) In any event, there is an elementary computable rule that tells us, for any Rd, which primes p split
and which remain prime in this agreed sense. The rule depends upon the residue of p modulo 4d: the reader is invited to guess it for the ring of Gauss- ian integers given the data just displayed above. In general, an elementary computable rule that says which primes split and which do not in a ring of algebraic integers such as Rd is referred to as a splitting law for the ring of algebraic integers in question.
5 The Rings Rd of Quadratic
Integers
There is a very important “symmetry,” or auto- morphism, defined on the ring Rd. It sends
d to −
d, keeps all ordinary integers fixed, and more generally, for rational numbers u and v, it sends α = u + v
d to what we may call its algebraic conjugate α′^ = u − v
d. (The word “algebraic” is to remind you that this is not necessarily the same as the complex-conjugate symmetry of the complex numbers!) You can immediately work out the formulas for this algebraic conjugation operation on the funda- mental quadratic irrationalities τd: if d is not con- gruent to 1 modulo 4, then τd =
d, so obviously τ (^) d′ = −τd, while if d is congruent to 1 modulo 4, then τd = 12 (1 +
d) and τ (^) d′ = 12 (1 −
d) = 1 − τd. This symmetry α → α′^ respects all algebraic for- mulas. For example, to work out the algebraic con- jugate of a polynomial expression like αβ + 2γ^2 , where α, β, and γ are numbers in Rd, you just replace each individual number by its algebraic conjugate, obtaining the expression α′β′^ + 2γ′^2. The most telling integer quantity attached to a number α = x + yτd in Rd is its norm N (α), which is defined to be the product αα′. This equals x^2 − dy^2 when τd =
d and x^2 + xy − 14 (d − 1)y^2 when τd = 12 (1 +
d). The norm turns out to be multiplicative, meaning that N (αβ) = N (α)N (β), as you can directly check by multiplying out the formula for the norm of each factor and comparing with the norm of the product. This gives us a use- ful tactic for trying to factorize algebraic numbers in Rd, and offers criteria for determining whether a number α in Rd is a unit, and whether it is prime in Rd. In fact, an element α ∈ Rd is a unit if and only if N (α) = αα′^ = ±1; in other words, the units
X + Y and keeping Y fixed. If we do this reversal and perform the corresponding simplification then we get back our original binary quadratic form. Because of this reversibility, these two quadratic forms take exactly the same set of integer values as X and Y vary: it is therefore reasonable to think of them as equivalent. More generally, then, one says that two binary quadratic forms are equivalent if one can be turned into the other (or minus the other) by any “reversible” linear change of variables with integer coefficients. That is, one chooses integers r, s, u, v such that rv − su = ±1, replaces X and Y by the linear combinations X′^ = rX +sY , Y ′^ = uX +vY , and simplifies the resulting expression to get a new triplet of coefficients. The condition rv − su = ± 1 guarantees that by a similar operation we can get back to our original binary quadratic form, and also that the new binary quadratic form has the same discriminant D as the old one. So when we talk of “essentially different” binary quadratic forms of discriminant D we mean that we cannot turn one into the other by this kind of change of variables. Here is the surprising obstruction to unique fac- torization that Gauss discovered.
The unique factorization principle is valid in Rd if and only if every homogeneous quadratic form aX^2 +bXY +cY 2 with dis- criminant equal to the fundamental dis- criminant of Rd is equivalent to the fun- damental quadratic form of Rd.
Furthermore, the collection of inequivalent quad- ratic forms whose discriminant is the fundamental discriminant of Rd expresses in concrete terms the degree to which Rd “enjoys unique factorization.” If you have never seen this theory of binary quadratic forms before, try your hand at working with quadratic forms in the case where D = −23. The idea is to start with some particular quad- ratic form aX^2 + bXY + cY 2 of your choice with discriminant D = b^2 − 4 ac = −23. Then, using a sequence of carefully chosen linear changes of vari- ables you reduce the size of the coefficients a, b, c until you can go no further. Eventually you should end up with one of the two (inequivalent) quad- ratic forms that there are with discriminant −23: the fundamental form X^2 + XY + 6Y 2 , or the form 2 X^2 +XY +3Y 2. For example, can you see that the
binary quadratic form X^2 + 3XY + 8Y 2 is equiv- alent to X^2 + XY + 6Y 2? This type of exercise offers a small hint of the role that the geometry of numbers will play in the eventual theory. As you might expect from the ven- erability of these ideas, elegant streamlined meth- ods have been discovered for making such calcula- tions. Nevertheless, it is an open secret that any working mathematician, contemporary or ancient, engaged in this subject or nearby subjects, has done a myriad of straightforward simple hand com- putations along the lines of the above exercise. If you try a few examples of this exercise, as I hope you do, here is one way of organizing your calculations. First, find a simple reversible linear change of variables to turn your form into an equiv- alent one with a, b, c 0. (You may also have to multiply the whole form by −1.) The cleanest way of writing down all binary quadratic forms given by triplets (a, b, c) of dis- criminant −23 is to list the triplets in increasing order of b, which will now be an odd positive inte- ger. For each value of b you can then choose a and c in such a way that their product is 14 (b^2 + 23). At this point the aim is to build up a repertoire of moves that tend to decrease b (which will keep a and c within bounds as well). A big clue, and aid, here is that for any pair of relatively prime integers x, y if you evaluate your quadratic form aX^2 +bXY +cY 2 at (X, Y ) = (x, y) to get the inte- ger a′^ = ax^2 +bxy +cy^2 , you can find, for appropri- ate b′^ and c′, a quadratic form a′X^2 + b′XY + c′Y 2 equivalent to yours, with first coefficient a′. So, one tactic is to look for small integers represented by your quadratic form. Also the “example” lin- ear change of variables X → X − Y , Y → Y will lead you to be able to reduce the coefficient b to an integer smaller than 2a. Can you check that X^2 + XY + 6Y 2 and 2X^2 + XY + 3Y 2 are inequiv- alent? Now, as we have just discussed, it follows from the general theory that R− 23 does not have the unique factorization property. We can also see this directly. For example,
τ− 23 · τ (^) −′ 23 = 2 · 3 ,
and all four of the factors in this equation are irre- ducible in R− 23. To be a faithful companion, I should at this point give at least a hint at what connection there might be between this specific
“failure of unique factorization” and the previous discussion. It may become a bit clearer in the next paragraph, but the underlying tension in the equa- tion τ− 23 · τ (^) −′ 23 = 2 · 3 is that all the factors in our ring are prime: we are missing any elements in our ring R− 23 that could factorize it further. We lack, for example, elements that play the role of the greatest common divisor of factors of this equation. The general theory regarding these mat- ters (which we are not entering into here, but see Euclid’s Algorithm) tells us that what is miss- ing is some element γ in R− 23 that is both a linear combination of the numbers τ− 23 and 2 (with coef- ficients in the ring R− 23 ) and also a common divi- sor of τ− 23 and 2 in the ring R− 23 , i.e. such that τ− 23 /γ and 2/γ are both in R− 23. There is no such element, for its norm must divide N (τ− 23 ) = 6 and N (2) = 4, and therefore be equal to 2, which can easily be shown to be impossible. But we are inter- ested, rather, in the phenomenon that inequiva- lence of certain binary quadratic forms will indeed show this, so let us go on. First, check that any linear combination α · τ− 23 + β · 2
with α, β elements of R 23 can also be written as u · τ− 23 + v · 2, where u and v are ordinary inte- gers. Now compute the binary quadratic form given by systematically taking the norms of these linear combinations, and viewing these norms as func- tions of the integer coefficients u, v:
N (u · τ− 23 + v · 2) = (τ− 23 u + 2v)(τ (^) −′ 23 u + 2v) = 6u^2 + 2uv + 4v^2.
Viewing the u and the v as variables, and dubbing them U and V to emphasize their status as vari- ables, we can say that the norm quadratic form obtained from the collection of linear combinations of τ− 23 and 2 is
6 U 2 + 2U V + 4V 2 = 2 · (3U 2 + U V + 2V 2 ). Now suppose that, contrary to fact, there were a common divisor, γ, as above; in particular, the multiples of γ in the ring R− 23 would then be pre- cisely the linear combinations of the numbers τ− 23 and 2. We would then have another way of describ- ing those linear combinations; namely, for any pair of ordinary integers (u, v) there would be a pair of ordinary integers (r, s) such that
u · τ− 23 + v · 2 = γ · (rτ− 23 + s) = rγτ− 23 + sγ.
Taking norms, as above, we would get
N (γ · (rτ− 23 + s)) = N (rγτ− 23 + sγ) = N (γ)(6r^2 + rs + s^2 ).
Again, thinking of r and s as variables and renam- ing them R and S we would have the corresponding norm quadratic form:
N (γ) · (6R^2 + RS + S^2 ) = 2 · (6R^2 + RS + S^2 ).
Given the above facts—dependent, of course, on the contrary-to-fact hypothesis that there is a γ as above—the key idea is that there would be linear changes of variables from (U, V ) to (R, S) and back that would establish an equivalence between the two quadratic forms 2 · (3U 2 + U V + 2V 2 ) and 2 · (6R^2 + RS + S^2 ). But these quadratic forms are not equivalent! Their inequivalence therefore shows that the putative γ does not exist and factorization in the ring R− 23 is not unique.
7 Class Numbers, and the Unique
Factorization Property
In the previous section we saw that the collection of inequivalent quadratic forms of discriminant equal to the fundamental discriminant provides us with an obstruction to unique factorization. Somewhat later, a more articulated version of this obstruc- tion arose, known as the ideal class group Hd of Rd. As its name implies, to describe this we must use the vocabulary of ideals and groups. For a gen- eral discussion of these concepts, see fundamen- tal concepts, groups and rings and ideals. A subset I of Rd is an ideal if it has the following closure properties: if α belongs to I, so do −α and τdα, and if α and β belong to I, so does α + β. (The first and third properties imply together that any integer combination of α and β belongs to I.) The basic example of such an ideal is the set of all multiples of some fixed, nonzero element γ of Rd, where by a multiple of γ we mean the product of γ and an element of Rd. We denote this set tersely as (γ), or, slightly more expressively, as γ · Rd. An ideal of this sort, i.e. one that can be expressed as the set of all multiples of a single nonzero ele- ment γ, is called a principal ideal. For example, the ring Rd itself is an ideal (it consists, after all, of all linear combinations of 1 and τd) and is even a principal ideal: in our laconic terminology, it can
the norm function on the elements of I, that is,
N (xα + yβ) = (xα + yβ)(xα′^ + yβ′) = αα′x^2 + (αβ′^ + α′β)xy + ββ′y^2.
This is a binary quadratic form in the variable coefficients x and y. If you start with a different choice of α, β that generate I you get a different form, but the two forms are scalar multiples of two forms with discriminant D that are equivalent to one another. Even better, the equivalence class of these forms depends only on the ideal class of I. It can be shown that there are only a finite num- ber of distinct ideal classes of Rd; that is, the ideal class group Hd is finite. The number of its elements is denoted hd and called the class number of Rd. So, the obstruction to unique factorization of Rd is given by the nontriviality of the group Hd; equiv- alently, unique factorization holds for Rd if and only if its class number is 1. But whether or not Hd is trivial, its detailed group-theoretic structure is profoundly related to the arithmetic of Rd. The class number enters into the generalizations of formulas (1.5) and (2.5) of Section 1; that is, the analytic formulas we alluded to in that sec- tion. These formulas represent just the beginning of one of the ongoing chapters of our subject, and form a bridge between the world of discrete arith- metical issues and that of calculus, infinite series, volumes of spaces, all of which can be attacked by the methods of complex analysis (see fundamen- tal concepts, holomorphic functions). Here is a sample of them.
(1) If d > 0 is a square-free integer and D is either d or 4d according to whether d is congruent to 1 modulo 4 or not, then
hd ·
log εd √ D
n 0
n
where the integers n run through those that are relatively prime to D and the signs ± are chosen in a way that depends only on the residue class of n modulo D.
(2) If d < 0 we have a somewhat simpler formula: there is no fundamental unit εd in Rd to con- tend with, but when d = −1 or −3, there are more roots of unity than merely ±1. If wd denotes the number of roots of unity in Rd, then w− 1 = 4, w− 3 = 6 and otherwise wd = 2,
and then one has a formula of the following type: hd wd
D
n 0
n
As d tends to −∞ the class number hd tends to infinity.
We have effective lower bounds for the growth of hd but these lower bounds are probably still far from the actual growth (cf. Goldfeld 1985). The effec- tive lower bounds that are known are exceedingly weak. They follow, however, from beautiful work of Goldfeld, and Gross and Zagier: for every real number r < 1 there is a computable constant C(r) such that hd > C(r) log |D|r^. Here is a sample:
hd >
p|D
p p + 1
· log |D|
if (D, 5077) = 1. It is a striking lacuna in our theory that, even today, nobody knows how to prove that there are infinitely many values of d > 0 for which Rd enjoys the unique factorization property— particularly since we expect that more than three- quarters of them do! Our expectations are even more precise than that, thanks to Henri Cohen and Hendrik Lenstra, who make use of certain prob- abilistic expectations (now known as the Cohen– Lenstra heuristics) to conjecture that the density of positive fundamental discriminants of class num- ber 1 among all positive fundamental discriminants is 0. 75446....
8 The Elliptic Modular Function
and the Unique Factorization
Property
A different obstruction to unique factorization in Rd is available when d is negative. Now Rd may be thought of as a lattice in the complex plane (see Figure 1.4), which makes a wonderful tool avail- able for us: the classical elliptic modular function of Klein,
j(z) = e−^2 πiz^ + 744 + 1 986 884 e^2 πiz
- 21 493 760 e^4 πiz^ + 864 299 970 e^6 πiz^ + · · ·. (8.1)
This function, also colloquially referred to as the “j-function,” converges for complex numbers z = x + iy with y > 0. If z = x + iy and z′^ = x′^ + iy′ are two such complex numbers, then j(z) = j(z′) if and only if the lattice generated by z and 1 in the complex plane is the same as the lattice generated by z′^ and 1 (or, equivalently, z′^ = (az +b)/(cz +d), where a, b, c, d are ordinary integers such that ad−bc = 1). We can paraphrase this by saying that the value j(z) depends only on, and characterizes, the lattice generated by z and 1. It turns out (by a theorem of Schneider) that if an algebraic number α = x + iy with y > 0 has the property that j(α) is also algebraic, then α is a (complex) quadratic irrationality; and the converse is also true. In particular, since α = τd is such a complex quadratic irrationality when d is negative, we have that the value, j(τd), of the j-function on τd is an algebraic number—in fact, an algebraic integer. This will be of some importance for our story. First, since the ring Rd as situated in the complex plane is simply the lattice generated by τd and 1, it follows from the previous paragraph that this value j(τd) will be the same if we replace τd by any element α of Rd, as long as the lattice generated by α and 1 is the entire ring Rd. More importantly, j(τd) is an algebraic integer of degree roughly comparable with the class number of Rd. In particular, it is an ordinary integer if and only if the ring Rd has the unique factorization property. (This result is one of the great applications of a classical theory known as complex multiplication.) In brief, here is yet another answer to the question of when the unique factorization principle holds for Rd when d is negative: if j(τd) is an ordinary integer, the answer is yes; otherwise it is no. The search for the full list of negative values of d for which Rd has the unique factorization property makes a marvelous tale: there are precisely nine values of d for which it occurs (see below), but for over two decades number theorists, while know- ing these nine, could prove only that there were no more than ten. The history of how the nonexistence of a possible tenth value of d was established, and reestablished, is one of the thrilling chapters in our subject. K. Heegner, in an article published in 1934, provided what he claimed was a proof of the nonexistence of the possible tenth value of d. However, Heegner’s proof was framed in somewhat unfamiliar language and was not understood by the
mathematicians of the time. His paper and his pur- ported proof were largely forgotten until the late 1960s, when the nonexistence of the tenth field was established (to the mathematical community’s sat- isfaction) by Stark (1967) and independently, via a different method, by Baker (1971). It was only then that mathematicians took a second and closer look at Heegner’s original article and discovered that he had indeed proven exactly what he claimed. More- over, his proof offered an elegant direct conceptual road to an understanding of the underlying issue. Here are the nine values of d:
d = − 1 , − 2 , − 3 , − 7 , − 11 , − 19 , − 43 , − 67 , − 163.
And here are the corresponding nine values of j(τd):
j(τd) = 2^633 , 2653 , 0 , − 3353 , − 215 , − 21533 , − 2183353 , − 2153353113 , − 2183353233293.
As Stark once pointed out, if, for some of these values of d, you simply “plug” τd into the power series expansion for j, you get rather surprising formulas. For example, when d = −163, then
e−^2 πiτd^ = −eπ
√ 163
is the first term of the power series for j(τ− 163 ) (see formula (8.1)). Since j(τ− 163 ) = − 2183353233293 and since all the terms e^2 πnτd^ (n > 0) that appear in the power series for the j-function are relatively small, we find that eπ
√ (^163) is incredibly close to an integer. Indeed, it is 2^183353233293 + 744 + · · · , which works out as 262 537 412 640 768 744 − , where the error term is less than 7. 5 × 10 −^13.
9 Representations of Prime
Numbers by Binary Quadratic
Forms
It happens more often than you might at first expect that difficult and/or somewhat artificial problems about ordinary integers can be translated into natural and tractable problems about larger rings of algebraic integers. My favorite elementary example of this type is the theorem due to Fermat that if a prime number p may be expressed as a sum of two squares, p = a^2 + b^2 with 0 < a b, then it has only one such expression. (For exam- ple, 1^2 +10^2 is the only way of expressing the prime
ican Mathematical Monthly.
11 Algebraic Numbers and
Algebraic Integers
Now that we have seen the algebraic integers j(τd) for negative values of d, and have touched on trigonometric sums, we have a few hints that, as with ordinary integers, the deep structure of these rings of quadratic integers may be better under- stood within a larger context of algebraic numbers. So now let us deal with algebraic numbers in full generality. By a monic polynomial, we mean a polynomial of the form
P (X) = Xn^ + a 1 Xn−^1 + · · · + an− 1 X + an,
i.e. a polynomial of degree n such that the coef- ficient of Xn^ is 1. In general, the other coeffi- cients are just assumed to be complex numbers. If P (X) = Xn^ + a 1 Xn−^1 + · · · + an− 1 X + an is such a polynomial, and if Θ is a complex number such that P (Θ) = 0, or, equivalently, if Θ satisfies the polynomial equation
Θn^ + a 1 Θn−^1 + · · · + an− 1 Θ + an = 0,
we say that Θ is a root of the polynomial P (X). The fundamental theorem of algebra, ini- tially proved by Gauss, guarantees that any such polynomial of degree n factors into a product of n linear polynomials. That is,
P (X) = (X − Θ 1 )(X − Θ 2 ) · · · (X − Θn)
for some complex numbers Θ 1 , Θ 2 ,... , Θn that are in fact precisely the roots of the polynomial P (X). If Θ is a root of such a polynomial P (X) = Xn^ + a 1 Xn−^1 + · · · + an− 1 X + an and if in addition the coefficients ai are rational numbers, then Θ is called an algebraic number. If the coefficients are not just rational but are in fact integers, then Θ is called an algebraic integer. So, for example, the square root of any rational number is an algebraic number and the square root of any “ordinary” inte- ger is an algebraic integer. The same holds true for nth roots of ordinary integers, or of algebraic inte- gers, for any natural number n. For an example of a different sort, we have already mentioned the the- orem that the values of the j-function on complex quadratic irrational integers are algebraic integers.
For a (random) particular case of that theorem, the complex number j(τ− 23 ) is a root of the monic polynomial
X^3 + 3 491 750X^2 − 5 151 296 875X
An exercise: show that any algebraic number can be expressed as an algebraic integer divided by an ordinary integer.
12 Presentation of Algebraic
Numbers
In dealing with any mathematical concept, we con- front, in one way or another, the dual problem of the various forms in which it comes to us when it arises in our work, and the various ways we can present it so as to deal with it effectively. We have already seen a bit of this at the outset of this arti- cle, in our discussion of quadratic surds, and we will continue to see it in our treatment of them below, where the various modes in which quadratic surds can be presented—as radicals, as eventually recurrent continued fractions, or as trigonometric sums—come together, all contributing to their uni- fied theory. This issue of presentation is all the more of a problem with algebraic numbers in general, which may come to us in a multitude of ways. For exam- ple, they can arise as the coordinates of points on specific algebraic varieties whose defining equa- tions may not be easily available, or as special values of functions like the j-function. It is nat- ural, then, to look for some uniform way of pre- senting algebraic numbers, and the history of the subject shows how much effort has been devoted to such a search. For example, consider the focus on iterated radical expressions, as in the famous for- mula for the solution to the general cubic equation X^3 = bX + c given by
X =
c 2
c^2 2
b^3 27
c 2
c^2 2
b^3 27
or the corresponding general solution to the fourth- degree equation. These were major achievements of sixteenth century Italian algebra, and they cul- minated in the proof that the general fifth-degree algebraic number could not be so expressed, which
was a major achievement of the early nineteenth century (see insolubility of the quintic). The challenge to give some analytic expression for such fifth-degree algebraic numbers was the source of a classic book by Klein, The Icosahedron, written in the late nineteenth century. Kronecker wrote that it was the “dream of his youth” (his Jugendtraum) to establish a uniform mode of presentation for a class of algebraic numbers that interested him, by expressing them as values of certain analytic func- tions.
13 Roots of Unity
A central role in the theory of algebraic numbers is played by the roots of unity, that is, the n complex solutions of the equation Xn^ = 1, or equivalently the n roots of the polynomial Xn^ − 1. If we let ζn = e^2 πi/n, then these roots are precisely ζn and its powers, so in particular they are algebraic inte- gers. They give us the factorization
Xn^ − 1 = (X − 1)(X − ζn)(X − ζ n^2 ) · · · (X − ζ nn− 1 ).
Now the powers of ζn form the vertices of a reg- ular n-gon in the complex plane, centered at the origin. This has the following consequence, noticed by Gauss in his youth. It can be shown that com- pass and straightedge constructions allow us, in effect, to extract square roots, so whenever ζn can be given as an expression built out of just square roots and the usual arithmetical operations, we have, implicitly, a ruler-and-compass construction of the regular n-gon, and conversely. To get some idea of why square roots are so closely connected with these constructions, con- sider this. If we have given ourselves a unit mea- sure, which we can view as the distance between the numbers 0 and 1 in the (complex) plane, and if we have already constructed, by whatever device, a specific point, x say, between 0 and 1 on the hor- izontal axis of the plane, we can first “construct” x/2 by straightedge and compass, and then go on to form a right-angled triangle with hypotenuse of length 1 + x/2 and one of its other sides of length 1 − x/2 (again using a straightedge and compass). Pythagoras’s theorem gives us that the third side of that triangle is of length
x. If one follows this line of thought (but adapts it to deal with complex quantities as well as the real number x as in the
example we have just discussed), then one can see that the equations
ζ 3 = 12 (1 + i
3), ζ 4 =
i,
ζ 5 = 14 (
5 − 1) + i (^18)
ζ 6 = − 12 (1 + i
provide (implicit) constructions of the equilateral triangle, the square, the regular pentagon, and the regular hexagon, respectively. By contrast, ζ 7 cannot be expressed solely in terms of the arith- metical operations and square roots (it is the root of a quadratic equation with coefficients that are rational expressions in the roots of the irreducible cubic polynomial X^3 − 73 X + 277 ), which already suggests that the regular heptagon might fail to be constructible by the standard classical means—and indeed it does fail without some act of “angle tri- section.” (In principle, though, the reader can work out an expression for ζ 7 in terms of square roots and cube roots by means of the information pro- vided in the parenthetical phrase above, together with equation (12.1).) Gauss showed that if n > 2 is a prime number then the regular n-gon is classically constructible if and only if n is a Fermat prime, that is, a prime number of the form 2^2
a
- So, for example, the 11-gon and 13-gon are not constructible by clas- sical means, but since ζ 17 is expressible as nested rational expressions of square roots, the 17-gon is, famously, constructible. So, not all roots of unity can be expressed as iterated rational expressions of square roots. How- ever, this inhospitability is not mutual, since all square roots of integers can be expressed as inte- ger combinations of roots of unity. More mysteri- ously, the elusive fundamental units εd (for d pos- itive), for which there is no known formula, are intimately related to a unit cd in Rd which is an explicit rational expression of roots of unity. (See below: it is called a circular unit.) This satisfies the elegant formula cd = εh d d, (13.1) which establishes yet another explicit test of unique factorization: the equality cd = εd is a “litmus” requirement for the unique factorization principle to hold in Rd. To give the flavor of the formulas involved, let p be an odd prime number and let a be an integer
which is of degree φ(p) = p − 1.
15 Algebraic Numbers as Ciphers
Determined by their Minimal
Polynomials
We have expressly insisted that our algebraic num- bers are complex numbers (of a certain sort). But another possible attitude towards an alge- braic number, Θ, an attitude at times promoted by Kronecker among others, is to deal with Θ as an unknown satisfying only the algebraic relations implied by the fact that it is a root of its (unique monic) minimal polynomial with rational coeffi- cients. For example, if the minimal polynomial of Θ is P (X) = X^3 − X − 1, then, according to this view, Θ is just an algebraic symbol that comes with the rule that any occurrence of Θ^3 may be replaced by Θ + 1 (rather as the complex number i can be regarded as a symbol with the property that i^2 may be replaced by −1). Any root of the minimal polynomial of Θ satisfies all the same polynomial relations with rational coefficients that Θ satisfies; these roots are called conjugates of Θ. If Θ is an algebraic number of degree n, then Θ has n distinct conjugates, all of them again, of course, algebraic numbers.
16 A Few Remarks About the
Theory of Polynomials
Central to the theory of polynomials in one variable—and, therefore, particularly to the theory of algebraic numbers—is the general relationship that roots have to coefficients:
∏^ n
i=
(X − Ti)
= Xn^ +
n∑− 1
j=
(−1)j^ Aj (T 1 , T 2 ,... , Tn)Xn−j^.
The polynomial Aj (T 1 , T 2 ,... , Tn) is homogeneous of degree j (this means that every monomial in it has total degree j), has integer coefficients, and is symmetric in (i.e. unchanged by any permutation of) the variables T 1 , T 2 ,... , Tn. The constant term is the product of the roots: An(T 1 , T 2 ,... , Tn) = T 1 · T 2 · · · · · Tn,
which is known as the norm form. The coefficient of Xn−^1 is the sum of the roots:
A 1 (T 1 , T 2 ,... , Tn) = T 1 + T 2 + · · · + Tn,
and this is the trace form. When n = 2 the norm and trace are all the sym- metric polynomials in the list. For n = 3, beyond the norm and trace we also have the symmetric polynomial of degree two:
A 2 (T 1 , T 2 , T 3 ) = T 1 T 2 + T 2 T 3 + T 3 T 1 = 12 {(T 1 + T 2 + T 3 )^2 − (T 12 + T 22 + T 32 )}.
It is of major importance to this theory, and more specifically to Galois theory, that the sym- metry properties of the conjugate roots are nicely reflected in these symmetric polynomials. In par- ticular, we have the fundamental result that any symmetric polynomial in T 1 , T 2 ,... , Tn with rational coefficients can be expressed as a poly- nomial with rational coefficients in the symmetric polynomials Aj (T 1 , T 2 ,... , Tn), and similarly with integral coefficients. For example, the equation above shows that T 12 + T 22 + T 32 can be expressed as A 1 (T 1 , T 2 , T 3 )^2 − 2 A 2 (T 1 , T 2 , T 3 ).
17 Fields of Algebraic Numbers,
Rings of Algebraic Integers
The inverse of a nonzero algebraic number is again an algebraic number; the sum, difference, and product of two algebraic numbers are algebraic numbers; the sum, difference, and product of two algebraic integers are algebraic integers. The neat proofs of these (latter) facts are a good demon- stration of the power of linear algebra, and in particular of Cramer’s rule. This states that any matrix with integer coefficients (and therefore also any linear transformation of a finite-dimensional vector space that preserves an integer lattice) satis- fies a monic polynomial identity with integer coef- ficients. To see just how useful this remark is for find- ing polynomial relations, and more specifically for showing that the collection of algebraic numbers and algebraic integers are closed under sums and products, try your hand at showing that
is an algebraic integer. One way to do it is to search for the monic fourth-degree polynomial equation that it satisfies. But this is hardly a beautiful cal- culation! If, however, you are familiar with lin- ear algebra, then a less painful route is to form the four-dimensional vector space over the rational numbers, generated by 1,
3, and
6 (which are linearly independent when the scalars are rational). Multiplication by
3 defines a lin- ear transformation T of this vector space, and one can compute its characteristic polynomial P. The Cayley–Hamilton theorem says that P (T ) = 0, and this translates into the statement that
3 is a root of P. These “closure properties” we have just dis- cussed lead us to study, in complete generality, fields of algebraic numbers and rings of algebraic integers. A number field is a field that is generated (as a field) by finitely many algebraic numbers. A standard result tells us that any number field K can in fact be generated by a single carefully cho- sen algebraic number. The degree of this algebraic number equals the degree of K, which is defined to be the dimension of K when K is viewed as a vector space over the field Q of rational numbers. One of the main introductory observations of Galois the- ory is that if K is a number field of degree n, then there are exactly n distinct ring-homomorphisms (“imbeddings”) ι : K → C from K into the field of complex numbers. (This means that ι sends 1 to 1 and respects the addition and multiplication laws within K. That is, ι(x + y) = ι(x) + ι(y) and ι(x · y) = ι(x) · ι(y).) From these imbeddings, we can construct some very useful rational-valued functions on K. For any element x in K, we form the n complex numbers x 1 , x 2 ,... , xn that are the images of x under the n different imbeddings of K into C. We then let
aj (x) := Aj (x 1 , x 2 ,... , xn),
where Aj (X 1 , X 2 ,... , Xn) is the jth symmetric polynomial of Section 14 above. (Because the poly- nomials Aj are symmetric, we do not have to worry about the order of the images x 1 , x 2 ,... , xn in the above expression.) It is not immediately obvious that the values of aj are rational numbers, but there is a theorem that tells us this. If an algebraic number Θ in K generates K (as a field), then the rational numbers aj (Θ) are the coefficients of its minimal polynomial; in general
they are the coefficients of a power of its min- imal polynomial. The most prominent of these functions are the multiplicative function an(x) = x 1 · x 2 · · · · · xn, called the norm function, usually denoted x → NK/Q(x), and the additive function a 1 (x) = x 1 +x 2 +· · ·+xn, called the trace function, usually denoted x → traceK/Q(x). The trace function can be used to define a funda- mental symmetric bilinear form on the Q-vector space K,
〈x, y〉 := traceK/Q(x · y),
which turns out to be nondegenerate. This nonde- generacy, together with the fact that if x, y are both algebraic integers, then 〈x, y〉 is an ordinary integer, can be used to show that the ring O(K) of all algebraic integers in K is finitely generated as an additive group. More specifically, there is a basis of algebraic integers in K, that is, a finite set {Θ 1 , Θ 2 ,... , Θn}, such that any other algebraic integer in K can be expressed as an “ordinary” integer combination of the numbers Θi. Let us summarize this structure. The number field K is a finite-dimensional vector space over Q and comes equipped with a nondegenerate bilinear symmetric form (x, y) → 〈x, y〉, and also with a lattice O(K) ⊂ K. Moreover, the restriction of the bilinear form to O(K) takes on integral values. The discriminant of K, denoted D(K), is defined to be the determinant of the matrix whose ij-entry is 〈Θi, Θj 〉, for {Θ 1 , Θ 2 ,... , Θn} a basis of the lattice O(K); this determinant does not depend on the basis chosen. The discriminant represents important informa- tion about the number field K. For one thing, there is a natural generalization to any number field of the notions of splitting and ramification that we discussed for quadratic fields, and the prime divi- sors p of D(K) are precisely those prime numbers that ramify in the field extension K. By a theorem of Minkowski, the absolute value of the discrim- inant D(K) of a number field K of degree n is always greater than ( π 4
)n ·
nn n!
This is greater than 1 unless K is the field of rational numbers. It follows that any nontrivial extension of the field of rational numbers has some
since these Galois groups and their linear represen- tations hold the key to a very detailed understand- ing of number fields. In its modern dress, algebraic number theory is closely connected with what is often called arithmetic algebraic geometry. Kronecker’s dream of getting explicit control of a wealth of algebraic number theoretic material by expressing algebraic numbers in terms of natu- ral analytic functions has not yet been fully real- ized. Nevertheless, the scope of this dream (and, one might also add, the supply of natural ana- lytic and algebraic functions) has expanded sub- stantially: the full range of algebraic geometry and group representation theory is now being brought to bear on it. This is done, for example, by the Langlands program, which among other things works with objects known as Shimura varieties. On the one hand, these varieties have close connec- tions with the theory of group representations and classical algebraic geometry, which greatly helps us to understand them. On the other hand, they are a rich source of concrete linear representations of Galois groups of number fields. This program, one of the glories of current mathematics, will, I expect, make a terrific chapter for a Companion to Math- ematics volume to be written at the beginning of the next century.
Further reading
Texts
First, three classics that require a minimum of background.
Gauss, C. F. 1986. Disquisitiones Arithmeticae, English edn. Springer. Davenport, H. 1992. The Higher Arithmetic: An Intro- duction to the Theory of Numbers. Cambridge Uni- versity Press. Hardy, G. H. and E. M. Wright. 1979. Introduction to Number Theory. Oxford University Press. At a more advanced level, the following are extraordinary expository books.
Borevich, Z. I. and I. R. Shafarevich. 1966. Number Theory. Academic Press. Cohen, H. 1993. A Course in Computational Algebraic Number Theory. Springer. Cassels, J. and A. Fr¨ohlich. 1967. Algebraic Number Theory. Academic Press. Ireland, K. and M. Rosen. 1982. A Classical Introduc- tion to Modern Number Theory, 2nd edn. Springer. Serre, J.-P. 1973. A Course in Arithmetic. Springer.
Technical Articles and Books Baker, A. 1971. Imaginary quadratic fields with class number 2. Annals of Mathematics (2) 94:139–152. Brauer, R. 1950. On the Zeta-function of algebraic number fields. I. American Journal of Mathematics 69:243–250. Brauer, R. 1950. On the Zeta-function of algebraic number fields. II. American Journal of Mathemat- ics 72:739–746. Goldfeld, D. 1985. Gauss’s class number problem for imaginary quadratic fields. Bulletin of the American Mathematical Society 13:23–37. Gross, B. and D. Zagier. 1986. Heegner points and derivatives of L-series. Inventiones Mathematicae 84:225–320. Heegner, K. 1952. Diophantische Analysis und Modul- funktionen. Mathematische Zeitschrift 56:227–253. Hua, L.-K. 1942. On the least solution of Pell’s equa- tion. Bulletin of the American Mathematical Society 48:731–735. Lang, S. 1970. Algebraic Number Theory. Reading, MA: Addison-Wesley. Narkiewicz, W. 1973. Algebraic Numbers. Warsaw: Pol- ish Scientific Publishers. Siegel, C. L. 1935. Uber die Classenzahl quadratischer¨ Zahl¨orper. Acta Arithmetica 1:83–86. Stark, H. 1967. A complete determination of the com- plex quadratic fields of class-number one. Michigan Mathematical Journal 14:1–27.
