Graduate Texts in Mathematics 11 Editorial Board F W Gehring P R Halmas (Managing Editor) C C Moore Graduate Texts in Mathematics A Selection 60 ARNOLD Mathematical Methods in C1assical Mechanics 61 62 WHITEHEAD Elements of Homotopy Theory KARGAPOLOV/MERZIJAKOV Fundamentals of the Theory of Groups 63 BOLLABAs Graph Theory 64 EDWARDS Fourier Series Vol I 2nd ed 65 66 WELLS Differential Analysis on Complex Manifolds 2nd ed W ATERHOUSE Introduction to Affine Group Schemes 67 SERRE Local Fields 68 69 WEIDMANN Linear Operators in Hilbert Spaces LANG Cyclotomic Fields 11 70 71 MASSEY Singular Homology Theory FARKAS/KRA Riemann Surfaces 72 73 STILLWELL C1assical Topology and Combinatorial Group Theory HUNGERFORD Algebra 74 75 DAVENPORT Multiplicative Number Theory 2nd ed HOCHSCHILD Basic Theory of Aigebraic Groups and Lie Aigebras 76 IITAKA Aigebraic Geometry 77 79 HEcKE Lectures on the Theory of Aigebraic Numbers WALTERS An Introduction to Ergodic Theory 80 ROBINSON A Course in the Theory of Groups 81 82 FORSTER Lectures on Riemann Surfaces BOTT/Tu Differential Forms in Aigebraic Topology 83 84 WASHINGTON Introduction to Cyclotomic Fields IRELAND/RoSEN A C1assicallntroduction to Modern Number Theory 85 EDWARDS Fourier Series: Vol 11 2nd ed 86 87 VAN LINT Introduction to Coding Theory BROWN Cohomology of Groups 88 89 PIERCE Associative Aigebras LANG Introduction to Aigebraic and Abelian Functions 2nd ed 91 92 BEARDON On the Geometry of Discrete Groups DIESTEL Sequences and Series in Banach Spaces 93 94 DUBROVIN/FoMENKO/NoVIKOV Modern Geometry-Methods and Applications Vol I WARNER Foundations of Differentiable Manifolds and Lie Groups 95 96 SHIRYAYEV Probability, Statistics, and Random Processes CONWAY A Course in Functional Analysis 97 KOBLITZ Introduction to Elliptic Curves and Modular Forms 98 99 BRÖCKER/tom DIECK Representations of Compact Lie Groups GROVE/BENSON Finite Reflection Groups 2nd ed 100 101 BERG/CHRISTENSEN/RESSEL Harmonic Analysis on Semigroups: Theory of positive definite and related functions EDWARDS Galois Theory 102 106 VARADARAJAN Lie Groups, Lie Aigebras and Their Representations SILVERMAN The Arithmetic of Elliptic Curves 107 OLVER Applications of Lie Groups to Differential Equations Serge Lang Alge braic N um ber Theory Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Serge Lang Department of Mathematics Yale U niversity New Haven, CT 06520 U.S.A Editorial Board P R Halmos F W Gehring C.C Moore M anaging Editor Department of M.athematics U niversity of Santa Clara Santa Clara, CA 95053 U.S.A Department of Mathematics University of l\Iichigan Ann Arbor, MI 48109 U.S.A Department of Mathematics University of California at Berkeley Berkeley, CA 94720 U.S.A AM.S Classitications: 1065 1250 With Illustrations Library of Congress Cataloging in Publication Data Lang, Serge Aigebraic number theory (Graduate texts in mathematics; 110) Bibliography: p Includes index Aigebraic number theory Class field theory I Title H Series QA247.L32 1986 512'.74 86-6627 Originally published in 1970 Reading, Massachusetts © by Addison-Wesley Publishing Company, Inc., © 1986 by Springer-Verlag New York Inc Softcover reprint ofthe hardcover 1st edition 1986 All rights reserved No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag 175 Fifth Avenue New Y ork, New York 10010 U.S.A 654 ISBN 978-1-4684-0298-8 ISBN 978-1-4684-0296-4 (eBook) DOI 10.1007/978-1-4684-0296-4 Foreword The present book gives an exposition of the classical basic algebraic and analytic number theory and supersedes my Algebraic Numbers, including much more material, e.g the class field theory on which I make further comments at the appropriate place later For different points of view, the reader is encouraged to read the collection of papers from the Brighton Symposium (edited by Cassels-Frohlich), the Artin-Tate notes on class field theory, Weil's book on Basic Number Theory, Borevich-Shafarevich's Number Theory, and also older books like those of Weber, Hasse, Hecke, and Hilbert's Zahlbericht It seems that over the years, everything that has been done has proved useful, theoretically or as examples, for the further development of the theory Old, and seemingly isolated special cases have continuously acquired renewed significance, often after half a century or more The point of view taken here is principally global, and we deal with local fields only incidentally For a more complete treatment of these, cf Serre's book Corps Locaux There is much to be said for a direct global approach to number fields Stylistically, I have intermingled the ideal and idelic approaches without prejudice for either I also include two proofs of the functional equation for the zeta function, to acquaint the reader with different techniques (in some sense equivalent, but in another sense, suggestive of very different moods) Even though areader will prefer some techniques over alternative ones, it is important at least that he should be aware of all the possibilities New York June 1970 SERGE LANG Prerequisites Chapters I through VII are self-contained, assuming only elementary algebra, say at the level of Galois theory Some of the chapters on analytic number theory assume so me analysis Chapter XIV assumes Fourier analysis on locally compact groups Chapters XV through XVII assume only standard analytical facts (we even prove so me of them), except for one aUusion to the Plancherel formula in Chapter XVII In the course of the Brauer-Siegel theorem, we use the conductordiscriminant formula, for which we refer to Artin-Tate where a detailed proof is given At that point, the use of this theorem is highly technical, and is due to the fact that one does not know that the zeros of the zeta function don't occur in a smaU interval to the left of If one knew this, the proof would become only a page long, and the L-series would not be needed at aU We give Siegel's original proof for that in Chapter XIII My Algebra gives more than enough background for the present book In fact, Algebra already contains a good part of the theory of integral extensions, and valuation theory, redone here in Chapters land II Furthermore, Algebra also contains whatever will be needed of group representation theory, used in a couple of isolated instances for applications of the class field theory, or to the Brauer-Siegel theorem The word ring will always mean commutative ring without zero divisors and with unit element (unless otherwise specified) If K is a field, then K* denotes its multiplicative group, and K its algebraic closure Occasionally, a bar is also used to denote reduction modulo a prime ideal We use the and notation If I, gare two functions of a real variable, and g is always ~ 0, we write = O(g) if there exists a constant C > such that I/(x) I ~ Cg(x) for all sufficiently large x We write = o(g) if limz +oo/(x)/g(x) = O We write/"" g if lim", _",/(x)/g(x) = vii Contents Part One General Basic Theory CHAPTER I Algebraic In tegers Localization Integral closure Prime ideals Chinese remainder theorem Galois extensions Dedekind rings Discrete valuation rings Explicit factorization of a prime 11 12 18 22 27 CHAPTER II COInpletions Definitions and completions Polynomials in complete fields Some filtrations Unramified extensions Tamely ramified extensions 31 41 45 48 51 CHAPTER III The Different and Discriminant 57 Complementary modules The different and ramification The discriminant 62 64 IX x CONTENTS CHAPTER IV Cyclotomic Fields l Roots of unity 71 Quadratic fields Gauss sums Relations in ideal classes 76 82 96 CHAPTER V ParalIelotopes l The product formula 99 110 116 119 Lattice points in parallelotopes A volume computation Minkowski's constant CHAPTER VI The Ideal Function l Generalized ideal classes Lattice points in homogeneously expanding domains The number of ideals in a given class CHAPTER 123 128 129 VII Ideles and Adeles l Restricted direct products Adeles Ideles Generalized ideal class groups; relations with idele classes Embedding of in the idele classes Galois operation on ideles and idele classes k: CHAPTER 137 139 140 145 151 152 VIII Elementary Properties of the Zeta Function and L-series l Lemmas on Dirichlet series Zeta function of a number field The L-series Density of primes in arithmetic progressions 155 159 162 166 CONTENTS xi PartTwo Class Field Theory CHAPTER IX Norm Index Computations Algebraie preliminaries Exponential and logarithm functions The loeal norm index A theorem on units The global eyelie norm index Applieations CHAPTER 179 185 187 189 192 194 X The Artin Symbol, Reciproeity Law, and Class Field Theory Formalism of the Artin symbol Existenee of a eonduetor for the Artin symbol Class fields CHAPTER 197 200 206 XI The Existenee Theorem and Loeal Class Field Theory Reduetion to Kummer extensions Proof of the existenee theorem The eomplete splitting theorem Loeal class field theory and the ramifieation theorem The Hilbert class field and the prineipal ideal theorem Infinite divisibility of the universal norms CHAPTER 213 215 217 219 224 225 XII L-series Again The proper abelian L-series Artin (non-abelian) L-series Indueed eharaeters and L-series eontributions 229 232 236 338 [XVII, §4] EXPLICIT FORMULAS have only a discontinuity of first kind, and such that (B) There is a number b > such that F(x) and F'(x) are O(e-O/Hbllzl) for lxi -+ 00 Under these hypotheses, we see that for < a' < b our function 4> is O(ltl- 1) uniformly in -a' ~ (1 ~ a' and the preceding discussion applies for < a < a' < band a ~ l The integrals on the line + a and -a will be reduced to integrals on the line i Our final result will be: + Explicitformula Let F(x) satisfy conditions (A) and (B), and let 4> be its Mellin transform The sum L:4>(w) taken over the zeros w = ß i'Y of L(s) satisfying ~ ß ~ and I'YI < T tends to a limit as T tends to infinity, and this limit is + lim T co L: I'YI (w) = ~xfco F(x) (e z / -co + e-z / 2) dx + 2F(0) log A - L: l~ ~: [x(ptF(log Npn) + x(p)-nF(log Np-n)] ~.n p where Fv(x) = F(x)e-i",oZ ancl qv is a functional to be described in §5 If one takes for F the function F(x) = {~Z/2 if x if < or x > log Y < x < log y for some fixed number y > I, then one recovers a classical formula as in Ingham, Chapter IV We leave the exact statement to the reader Observe that conditions (A) and (B) are obviously satisfied, and that in the sum taken over p, n only the terms with positive powers of p will appear §4 Evaluation 01 the sum: First part In our explicit formula, the term with from the definition of 4>(0) and 4>(1) For the others, we use the identity cl log A = cl log Go ~x arises in the obvious fashion + cl log L + L: vES", d log Gv and compute our two integrals successively for the three cases [XVII, §4] EVALUATION OF THE SUM: FIRST PART 339 Take first the case of Go We have d dslog Go(s) = log A, d dslog Go(l - s) = -log A In view of the fact that we integrate a holomorphic function, we can shift both integrals to the line (T = i, and combine them into the integral log A 211"1, f 1/ + iT «II(s) ds 1/2-iT We make the substitution s = t + it, ds = idt, and take the limit for T -+ 00 The Fourier inversion formula is applicable, and we find the desired expression 2F(Q) log A as an answer We take up next the case of the integrals over d log L We look first at the integral on the line + a A trivial computation yields the value 'Ir1 J +T dt:E J+oo Htn(u)e itu du, -T -00 where Htn(u) = l~~~~ x(p)nF(u + log N pn)e'" [f_'"'" - e-~Izl Iez/2 _ e -z/21 ß(x) dx - ] 2ß(0) log> Of course, we must prove: Lemma exists On the space of almost BeL functions, the preceding limit Proof In a neighborhood of 0, the denominator lez / - e-z / behaves like lxi (mod x ) It will then suffice to prove our assertion when we replace this denominator by lxi and ß by a function which tends rapidly to at infinity Using linearity, consider first the case of a characteristic function If the origin does not lie in its interval, then the limit clearly exists If the origin lies in the interval, then we are led to consider an integral of type b > 0, >'!i; We can differentiate under the integral sign, and taking b plicity, we get -~ > = for sim- Hence ') = log> + an integral which converges for > -t 00 From this it is clear that the term 2ß(0) log> will cancel, and leave an integral whose limit exists as > -t 00 Next, suppose that ß is BeL and tends rapidly to at infinity Decomposing ß into its odd and even parts, and subtracting a characteristic function we may assurne that ß is even and ß(O) = o In that case, ß(x)/lxl is bounded in some neighborhood of the origin, and hence the integral f '" -'" (1 - e-~Izl) ß(x) dx lxi has a limit as > -t 00 The term involving 2ß(0) log> is 0, and so our assertion is proved in that case Let ß be a BeL function If we write ß(x) = ß(x) - ß(O) + ß(O), [XVII, §51 343 EVALUATION OF THE SUM: SECOND PART we obtain by linearity W(ß)'= ß(O)W(l) + f +OO _00 ß(x) - ß(O) d lez / _ e- z / 21 x There is no convergence problem about this last integral, and thus we find: Lemma If ß is a BGL function, then IW(ß) I ~ G(ißI + liPl ß), where G is a fixed constant, and li P ß is a Lipshitz constant on some compact interval containing O The above lemma allows us to prove a continuity property for our functional W, namely: Lemma Let {ßn} be a sequence of BGL functions, converging to a BGL function ß Assume also that the functions {ßn} are uniformly bounded, that the convergence is uniform on every compact set, and that the Lipshitz constants lip ßn are bounded on every compact set Then W(ßn) converges to W(ß) Proof We write for each n, Then ßn(O) converges to ß(O) This reduces the proof to considering the sum of the integrals over intervals lxi lxi ~ A lxi ~ E A ~ E ~ taking E > small and A large For A large, the exponential function in the denominator makes the integral small For E smalI, the last integral has a small value in view of the uniform bound for the Lipshitz constants The integral in the middle range is then close to the corresponding integral for ß(x) - ß(O) Thus our lemma is clear Let ß be as before If y is any number, we denote by ß1I the function given by ß1I (x) = ß(x + y) Then by what we said above, the function W(ß 1I ) is continuous (as a function of y) 344 EXPLICIT FORMULAS [XVII, §5] Let {Pn} be a sequence of regularizing functions, i.e infinitely differentiable ~ 0, with compact support shrinking to 0, and whose integral is equal to We can form the convolution ß * Pn, and it is easily verified that the sequence of functions {ß * Pn} converges to ß in the sense of the additional assumptions of Lemma for this convergence For any function ß, we denote by ß- the function Then we obtain: Lemma Let {Pn} be a regularizing family as above Then the functions W((ß * Pn)-x) converge to W(ß-x) uniformlyon every compact set From this we conclude: Lemma The functional W is a distribution Let ß be a BGL function The convolution of W with T ß (the distribution represented by ß) is representedbythefunction whose value at x is W(ß-x) Symbolically, This function is continuous Proof If T is a distribution, which is represented outside so me compact set by a function tending exponentially to at infinity, and a is a Goo_ function which is bounded, then by the theory of distributions, one knows that is represented by the function T(a- x ), which has a meaning in this case We can apply this result to the functions ß * Pn, and hence our lemma follows [Cf TD, Theorem XI of Chapter VI, §4 and formula (VI, 1; 2).] Weshall now prove the analogue of Lemma for almost BCL functions Lemma Let X be a characteristic function of an interval u'hich does not ho,ve as its endpoints Then the distribution W * T x is represented by a Goo-function locally at every point other than the endpoints of the interval, and its value at such a point x is the value W(x-x) Proof This follows from the general properties of convolutions of distributions, e.g TD, Chapter VI, Theorem III of §3 and Theorem XI of §4 Corollary Let F be a function which is almost BGL, and is continuous at O Then the distribution W * T F is represented by a Goo-function locally at every point other than the points of discontinuity of F, and the value of [XVII, §5] 345 EVALUATION OF THE SUM: SECOND PART this function at is We now come to more specific considerations concerning the gamma function From its Weierstrass product, one gets (in every book on the gamma function) r' jr(z) = - - - 'Y Z [1 1] + n=l ~ - - n n +z 00 ! + it and taking the real part, Substituting z = 'Y = lim M-+oo together with the limit (1 + + ~ - log M) M we obtain at once denoting by qM the function inside the brackets Lemma The convergence of thislirnit is uniform on every compact set Furthermore for some constant Cindependent of M, and say Itl ~ Proof The first assertion is clear As to the second, observe that the sum in the expression for qM is bounded from below by (n n+! + !)2 + t2 O For M ~ t the expression for qM is bounded by log M For M ~ t, we observe: M M n=O n=t ~ ~ ~ ~ log M - log t - constant, which gives us again what we want ~ log t 346 EXPLICIT FORMULAS [XVII, §5] For the rest of this section, we agree to normalize the Fourier transform as follows For suitable functions j, we define f +OO j(x)e-' J(t) = 't _00 x dx We also denote by ( , ) the integral of the product of the two functions appearing in the blank space of this scalar product, whenever the integral makes sense Then formally (for a restricted class of functions j, g), the Plancherel forrnula asserts that 0) Hence if we write qM = 10gM + YM, then we find In the sense of distributions, the Fourier transform of the constant function is f = 27ro, where obtain: IS the functional oU) = j(O) From these considerations, we Lemma 10 As a distribution, Proof The boundedness condition of Lemma ins ures that the limit of the Fourier transforms is the Fourier transform of the limit (Use TD, Example of Chapter VII, §7.) [XVII, §5] EVALUATION OF THE SUM: SECOND PART 347 Our goal is to prove: Proposition Let F be a Junction satisJying properties (A) and (B) Let t/I= Then P- .], where Kv(x) is the function given by the formulas: e(!-Imvlllxl Kv(x) = Ie X e- X I - if N v = e-!lmvxl Kv(x) = IeX I - if N v = e-X I I The same type of computation that gave us the Fourier transform of ll(t) gives us the Fourier transform of llv, and one obtains: Lemma 13 The Fourier transJorm oJ llv is given by ~ qv = - 27r Nv W v• Putting everything together, we obtain: Proposition The last sum in the explicit Jormula is equal to "L J N ~ v) = -2-v qv(F vES", 7r "L J vES", W v(F v) Bibliography [I] E ARTIN, Algebraic numbers and algebraic functions, Lecture notes by Adamson, Gordon and Breach, New York, 1967 [2] - - - , Theory of algebraic numbers, Notes by G Wurges, translated by G Striker, Göttingen, 1956 [3] E ARTIN and J TATE, Class Field Theory, Benjamin, New York, 1967 [4] Z Borevieh and Shafarevich, Number Theory, Academic Press, New York, 1966 [5] N BOURBAKI, Commutative Algebra, Hermann, Paris, 1962 [6] R BRAUER, "On the zeta-functions of algebraic number fields II," Am J lIfath 72 (1950), 739-746 [7] J W CASSELS and A FROHLICH, Algebraic Number Theory, Proceedings of the Brighton Conference, Academic Press, New York, 1968 [8] H HASSE, "Bericht uber Neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkorper," Jahresbericht D lIfath Ver., 1926, 1927, and 1930 [9] H HASSE, Vorlesungen uber Klassenkorpertheorie, Marburg notes, reprinted by Physica Verlag, 1967 [10] D HILBERT, "Die Theorie der algebraischen Zahlkörper," Jahresbericht D lIfath Ver (1897), 175-546 [11] E LANDAU, Einfuhrung in die elementare und analytische Theorie der algebraischen Zahlen und der ideale, Chelsea, New York, 1949 [12] J P Serre, Corps Locaux, Hermann, Paris, 1963 (13) J TATE, Fourier analysis in nttmber fields and Hecke's zeta function, Thesis, Princeton, 1950, also appears in (7) (14) A WEIL, Sur les "formttles explicites" de la theorie des nombres premiers, Comm Seminaire Math Universite de Lund (dedie a M Riesz) (1952) pp 252-265 (15) A WEIL, Basic Number Theory, Springer Verlag, New York, 1968 351 Index abscissa of convergence, 156 absolute value, 31 algebraic integers, archimedean absolute value, 35 Artin automorphism, 18 Artin kerneI, 197 Artin map, 197 Artin symbol, 197 associated ideal, 111 canonical set of absolute values, 35 Chinese Remainder Theorem, 11 dass field, 208, 212 dass group, 208 dass number, 100, 123 complementary set, 57 completion, 32, 36 complex absolute value, 35 conductor, 199, 229, 279 consistency of Artin map, 198 convex, 116 cyde, 123 cydotomic extension, 71 decomposition field, 13 decomposition group, 13 Dedekind ring, 20 dependent, 32 different, 60 Dirichlet density, 167 discrete valuation ring, 22 discriminant, 64 equidistributed, 128, 316 equivalent norms, 33 exponential function (p-adic), 185 finite part, 124 formal ideal, 40 fractional ideal, 18 Frobenius automorphism, 17 fundamental domain, 112, 128 fundamental units, 108 Gauss sums, 82 generalized ideal dasses, 145 Hecke character, 293 Hensel's Lemma, 43 Herbrand quotient, 179 Hilbert dass field, 224 ideal dass, 22, 123, 124 idele, 138 idele dass, 142 independent, 32 inertia field, 16 inertia group, 16 integers, integral,5 integral equation, integrally dosed, Krasner's lemma, 43 Kronecker's theorem, 210 Kummer extensions, 213 lie above, linearly equivalent ideals, 22 Lipschitz parametrizable, 128 local different, 62 local field, 92 local ring, 353 354 INDEX logarithm (p-adic), 185 L-series, 162, 229 quadratic symbol, 76, 81 quasi character, 278, 287 Minkowski constant, 120 M K ,35 lIh-divisor, 101 multiplicative character, 82 multiplicative subset, multiplicativity of different, 60 ramification index, 24, 279 ray class field, 209 real absolute value, 35 reciproeity law map, 197 regulator, 109, 127 residue class degree, 24 restrieted direet produet, 138 Nakayama's lemma, norm, 33 norm index, 164 norm of ideals, 24 norm of ideles, 138 number field, orbit of units, 130 order of element, 20 order of ideal, 20 p-adic integers, 37 p-adic number, 37 parallelotope, 112 p-component, 40 pole, 20 prime, 21 prime at infinity, 99 primitive character, 82 principal fractional ideal, 22 principal idele, 142 Q-machine, 181 quadratic reciprocity law, 77 self-dual Haar measure, 275 semiloeal, 183 S-idele, 138 S-idele class, 142 size, 101 split completely, 16, 25, 39 Stickelberger's eriterion, 67 strongly ramified, 52 S-unit, 104 symmetrie, 116 tamely ramified, 52 totally ramified, 51 units, 21, 104 universal norm, 212, 225 unramified,48, 147, 221, 278 valuation, 31 zero, 20 ... proves one inequality, and shows that the norm 11 is continuous with respect to 11 11 Hence 1I has a minimum on the unit sphere with respect to 11 11 (by local compactness), say at the point v E... since feX) - P1(x)e l ••• Pr(x)e r E 'pA [X], it folIo ws that On the other hand, we see that whence using (*) we find ~11 ? ?11 3: r C 'pB + P1(a)e l • •• I13r(a) erB C 'pB = I13r; ? ?11 3:~ This proves... and idele classes k: CHAPTER 13 7 13 9 14 0 14 5 15 1 15 2 VIII Elementary Properties of the Zeta Function and L-series l Lemmas on Dirichlet series Zeta function of a number field The L-series Density