DOCSLIB.ORG

1. Generalized Prime Number Theorem. by Takayoshi MITSUI

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
1. Generalized Prime Number Theorem. by Takayoshi MITSUI 1. Generalized Prime Number Theorem. By Takayoshi MITSUI (ReceivedJanuary 31, 1957) Let K be an algebraic number field of degree n, K(1),•c, K(ƒÁ1) the real conjugate fields of K, and K(ƒÁ1+1),•c, K(n) (n=ƒÁ1+2ƒÁ2) the complex conjugate fields of K where K(p+ƒÁ2)=(p=ƒÁ1+1,•c,ƒÁ1+ƒÁ2), we denote by P the direct product of ƒÁ1 real lines and ƒÁ2 complex planes, Let ƒÊ be an ideal number of K , and ƒÊ(1),•c,ƒÊ(n) the conjugates of ƒÊ as defined by Hecke [2] , Then V(ƒÊ)=(„ ƒÊ(1)„ ,•c,„ ƒÊ(ƒÁ1)„ ,ƒÊ(ƒÁ1+1),•c, ƒÊ(ƒÁ1+ƒÁ2) can be regarded as a point in P, and it is a problem to compute the number of prime ideal numbers ƒÖ such that V(ƒÖ) are in a certain domain in P. In case n+ƒÁ1=2, Rademacher [7] obtained the following result; let K be a real quadratic number field, Y1, Y2 positive number, a an ideal of K, ƒÏ a totally positive integer of K, (ƒÏ, a) t, If we denote by ƒÎ(ƒÏ, a; Y1, Y2) the number of prime numbers w satisfying the conditions (ƒÖ=ƒÏ(mod a), 0<ƒÖ(1)•…Y1, 0<ƒÖ(2)•…Y2, then we have (1) where h is the number of ideal classes, is a totally positive fundamental unit such that ƒÅ>1. On the other hand, it is known (Siegel [8], walfisz •m10]) that we have (2) where the remainder term is uniform with respect to the modulus k, that is , the constants in the remainder term do not depend on k under the assumption k•…(log x) A for a fixed positive number A. In the present paper, we shall obtain a result including all above theorems as special cases. Our Main Theorem will be stated and proved in our final paragraph •˜ 4, after some preparations in •˜•˜1-3. we begin by generalizing (2) to the case of arbitrary algebraic number field in •˜ 1. The aim of this paragraph will be attained by the theorem on ƒÎ(x, _??_), the number of prime ideals p in the class _??_ with Np•…x, given at the end of •˜1. Now ƒÎ(x, _??_) may be considered as a special case for ƒÉ=1 of a certain sum (see (3.52)) for Grossencharacter A of K. The estimation of this sum‡”*ƒÉ(ƒÖ) for the case ƒÉ=1 will be given as Lemma 8 in •˜2, after seven lemmas concerning the properties of ƒÄ-f unction with Grossencharacter, In •˜2, we shall prove three more lemmas needed to establish the uniformity of the remainder term of our final for 2 Takayoshi MITSUI mula with respect to the modulus. In •˜3, we shall generalize the following theorem of Rademacher [6] by means of which he could establish his above cited result (1); let K be a real quadratic field, a an ideal, ƒÅa a totally positive fundamental unit mod a such that and define a function W(ƒÊ) for ƒÊ=K as follows; If we take a totally positive integer o prime to a and denote by ƒÎ(a,ƒÏ; x, v) (0<v •…1) the number of totally positive prime numbers ƒÖ of K satisfying the conditions; ƒÖ=p (mod a), NƒÖ•…x, 0•…W(ƒÖ)<v, then we have The theorem, which will be the aim of this paragraph, will be formulated at the beginning. All results of •˜•˜1-2 are utilized in the proof of this theorem, by which our Main Theorem will be then proved in •˜4. The author wishes to express his cordial thanks to Professor S. Iyanaga for his encouragement during the preparation of this paper. 1. Improvement of prime ideal theorem. We consider Hecke's L-function L(s, x) of an algebraic field K with degree n, where x is a character mod a (a means the product of an ideal a and some infinite prime ideals p(i)•‡). We write as usual s=ƒÐ+it, ƒÐ=k(s), t=k(s) and denote by x0 the principal character and by A1, A2,•c positive absolute constants. (For details on the properties of L-functions, cf. Landau [3]). LEMMA 1. If x is a character mod a, then we have in the strip -1/2•…ƒÐ•…4 (1.1) „ L(s, x)(s-1)E(x)„ <A1(Na(1+„ t„ ))A2, where PROOF. It is obvious that L(s, x)(s-1)E(x) is regular in the strip -1/2•…ƒÐ•…4. First assume that x is a primitive character. On the line ƒÐ=4, we have „ L(s, x)(s-1)E(x)„ •…A3(1+„ t„ ), and on the line ƒÐ=-1/2, we have by the functional equation of L(s, x), „IL(s, x)(s-1)E(x)„ •…A4Na(1+„ t„ )n+1 Finally in the strip -1/2•…ƒÐ<4, we have„ L(s, x)(s-1)E(x)„ •…C1ec2t t, where C1, C2 are positive constants independent of t, ƒÐ. Hence by theorem of Phragmen-Lindelof, we have Generalized Prime Number Theorem. 3 (1.2) „ L(s, x)(s-1)E(x) „ •…A5Na(1+„ t„ )n+1. Next assume x is not primitive and let x0 be the primitive character induced from x. Then we have L(s, x)=L(s, x0) II (1-x0(p)Np-s), in which II(1-x0(p)Np-s)„ •…II(1+Np1/2)•…A6IINp•…A6Na. Therefore (1.1) is obtained. LEMMA 2. In the region „ t„ •†4,ƒÐ•†1-A7/log(„ t„ Na), we have L(s, x) •‚0. PROOF. 1t suffices to prove for t•†4. Suppose that ƒÀ+iƒÁ(ƒÁ•†4) is a zero point of L(s, x). Taking ƒÐ0=1+A/log(ƒÁNa) (A>0 will be determined later) we denote ƒÐ0+iƒÁ by s0. Then In the circle „ s-s0„ •…1/2, we have by Lemma 1 Therefore by lemma ƒÀ of Titchmarsh [9] p. 49, We obtain analogously Taking A sufficiently small, we have Hence, by a well-known inequality we obtain which gives Since we may take A small, the value of the formula in the brackets is surely positive, that is, Thus lemma is proved. LEMMA 3. (Titchmarsh) Suppose that f(s) is regular in the circle „ s-s0„ •… 4 Takayoshi MITSUI and „ f (s)/f(s0)„ <eM (M>1), „ f•L(s0)/f(s0)„ <M/r, further f(s)•‚0 in the domain defined by the conditions „ s-s0„ •…r and ƒÐ•†ƒÐ0-2ƒÁ•L (0<ƒÁ•L <ƒÁ/4). Then we have („ s-s0„ •…r). PROOF. See Titchmarsh [9] p. 50. LEMMA 4. In the region „ t„ •†5, ƒÐ•† 1-A7/8 log(„ t„ Na), we have (1.3) PROOF. Again we may assume t•†5. Let ƒÁ•†5 and put s0=ƒÐ0+iƒÁ. Then L(s, x) is regular in the circle „ s-s0„ •…1/2 and by Lemma 2 L(s, x)•‚0 in the domain defined by the conditions „ s-s0„ •…1/2 and ƒÐ•†ƒÐ0-2ƒÁ•L. Further we have in the circle„ s-s0„ •…1/2 and Therefore, putting A19=max(A17, A18/2), M=A19log(ƒÁNa), ƒÁ=1/2, we have by Lemma 3 („ s-s0„ •…ƒÁ). (1.3) follows at once. LEMMA 5. If x is a complex character mod a, then in the region „ t„ •…10, ƒÐ•†1 -A21/log(Na+1), L(s, x) has no zero point. PROOF. This is similar to the proof of Lemma 2, so we may omit it. LEMMA 6. If x is a real character mod a and x•‚x0, then L(s, x)•‚0 in the region; PROOF. Suppose that lemma is not true, then for any small positive number c there exists a zero point ƒÀ+iƒÁ of L(s, x) such that Now let a be a positive number such that 1•†((a+1)2+4a2)(a+1/2) and put ƒÐ0=1 +c/a log(Na+1), s0=ƒÐ0+iƒÁ. Taking c small, we may assume that two zero points ƒÀ+ iƒÁ, ƒÀ-iƒÁ of L(s, x) are in the circle „ s-s0„ •…1/4. In the circle „ s-s0„ •…1/2, L(s, x) is regular and „ L(s, x)„ <(Na+1)A24, so we have Generalized Prime Number Theorem. 5 („ s-s0„ •…1/2). Therefore by the lemma a of Titchmarsh [9] p. 49 (1.4) where ƒÏ runs through the zero points of L(s, x) in the circle „ s-s0•…1/4, so we have clearly Again taking c sufficiently small, we have so we have from (1.4) Further taking e small, This and the inequality give (1.5) This is a contradiction and proof is completed. LEMMA 7. If x is a real character mod a and x•‚x0, then L(s, x)•‚0 in the region 0<„ t„ •…6, ƒÐ•†1-A28/log(Na+1). PROOF. Let ƒÀ+iƒÁ (0<ƒÁ•…6) be a zero point of L(s, x). Then by Lemma 6, we may assume A22/log(Na+1)•…ƒÁ•…6. Put ƒÐ0=1+A/log(Na+1) (A>0) and consider (1.6) First taking A sufficiently small, we have Next in the circle „ s-s0„ •…1/2 (s0=ƒÐ0+iƒÁ) so we have Finally putting s0=ƒÐ0+2iƒÁ, we obtain 6 Takayoshi MITSUI in the circle •bs0-s•b•…1/2, hence by lemma ƒÀ of Titchmarsh [9] p. 49 Since A may be arbitrarily small, we may assume that A/(A2+A222)<1/4, that is, that the inequality holds.
Load more

Recommended publications
  • Methodology and Metaphysics in the Development of Dedekind's Theory
    Methodology and metaphysics in the development of Dedekind’s theory of ideals Jeremy Avigad 1 Introduction Philosophical concerns rarely force their way into the average mathematician’s workday. But, in extreme circumstances, fundamental questions can arise as to the legitimacy of a certain manner of proceeding, say, as to whether a particular object should be granted ontological status, or whether a certain conclusion is epistemologically warranted. There are then two distinct views as to the role that philosophy should play in such a situation. On the first view, the mathematician is called upon to turn to the counsel of philosophers, in much the same way as a nation considering an action of dubious international legality is called upon to turn to the United Nations for guidance. After due consideration of appropriate regulations and guidelines (and, possibly, debate between representatives of different philosophical fac- tions), the philosophers render a decision, by which the dutiful mathematician abides. Quine was famously critical of such dreams of a ‘first philosophy.’ At the oppos- ite extreme, our hypothetical mathematician answers only to the subject’s internal concerns, blithely or brashly indifferent to philosophical approval. What is at stake to our mathematician friend is whether the questionable practice provides a proper mathematical solution to the problem at hand, or an appropriate mathematical understanding; or, in pragmatic terms, whether it will make it past a journal referee. In short, mathematics is as mathematics does, and the philosopher’s task is simply to make sense of the subject as it evolves and certify practices that are already in place.
    [Show full text]
  • Arxiv:Math/0412262V2 [Math.NT] 8 Aug 2012 Etrgae Tgte Ihm)O Atnscnetr and Conjecture fie ‘Artin’S Number on of Cojoc Me) Domains’
    ARTIN’S PRIMITIVE ROOT CONJECTURE - a survey - PIETER MOREE (with contributions by A.C. Cojocaru, W. Gajda and H. Graves) To the memory of John L. Selfridge (1927-2010) Abstract. One of the first concepts one meets in elementary number theory is that of the multiplicative order. We give a survey of the lit- erature on this topic emphasizing the Artin primitive root conjecture (1927). The first part of the survey is intended for a rather general audience and rather colloquial, whereas the second part is intended for number theorists and ends with several open problems. The contribu- tions in the survey on ‘elliptic Artin’ are due to Alina Cojocaru. Woj- ciec Gajda wrote a section on ‘Artin for K-theory of number fields’, and Hester Graves (together with me) on ‘Artin’s conjecture and Euclidean domains’. Contents 1. Introduction 2 2. Naive heuristic approach 5 3. Algebraic number theory 5 3.1. Analytic algebraic number theory 6 4. Artin’s heuristic approach 8 5. Modified heuristic approach (`ala Artin) 9 6. Hooley’s work 10 6.1. Unconditional results 12 7. Probabilistic model 13 8. The indicator function 17 arXiv:math/0412262v2 [math.NT] 8 Aug 2012 8.1. The indicator function and probabilistic models 17 8.2. The indicator function in the function field setting 18 9. Some variations of Artin’s problem 20 9.1. Elliptic Artin (by A.C. Cojocaru) 20 9.2. Even order 22 9.3. Order in a prescribed arithmetic progression 24 9.4. Divisors of second order recurrences 25 9.5. Lenstra’s work 29 9.6.
    [Show full text]
  • A Brief History of Ring Theory
    A Brief History of Ring Theory by Kristen Pollock Abstract Algebra II, Math 442 Loyola College, Spring 2005 A Brief History of Ring Theory Kristen Pollock 2 1. Introduction In order to fully define and examine an abstract ring, this essay will follow a procedure that is unlike a typical algebra textbook. That is, rather than initially offering just definitions, relevant examples will first be supplied so that the origins of a ring and its components can be better understood. Of course, this is the path that history has taken so what better way to proceed? First, it is important to understand that the abstract ring concept emerged from not one, but two theories: commutative ring theory and noncommutative ring the- ory. These two theories originated in different problems, were developed by different people and flourished in different directions. Still, these theories have much in com- mon and together form the foundation of today's ring theory. Specifically, modern commutative ring theory has its roots in problems of algebraic number theory and algebraic geometry. On the other hand, noncommutative ring theory originated from an attempt to expand the complex numbers to a variety of hypercomplex number systems. 2. Noncommutative Rings We will begin with noncommutative ring theory and its main originating ex- ample: the quaternions. According to Israel Kleiner's article \The Genesis of the Abstract Ring Concept," [2]. these numbers, created by Hamilton in 1843, are of the form a + bi + cj + dk (a; b; c; d 2 R) where addition is through its components 2 2 2 and multiplication is subject to the relations i =pj = k = ijk = −1.
    [Show full text]
  • Appendix a Solutions and Hints to Exercises
    Appendix A Solutions and Hints to Exercises Chapter 1 1.1 1. Suppose that there are only finitely many prime p ≡ 3 (mod 4), and label them p1,...,pn. Then consider N = 4p1 ...pn − 1. This number is clearly odd, and has remainder 3 after dividing by 4. So its prime divisors are all odd, and they can’t all be 1 (mod 4), as the product of numbers which are 1 (mod 4) is again 1 (mod 4). So N has a prime divisor p ≡ 3(mod4).But p must be different from all the pi , since p|N but pi N for all i = 1,...,n. So there must be infinitely many primes p ≡ 3(mod4). 2. Suppose that there are only finitely many prime p ≡ 5 (mod 6), and label them p1,...,pn. Then consider N = 6p1 ...pn − 1. This number is clearly odd, and has remainder 5 after dividing by 6. So (6, N) = 1, and the prime divisors of N can’t all be 1 (mod 6), as the product of numbers which are 1 (mod 6) is again 1 (mod 6). So N has a prime divisor p ≡ 5(mod6).But p must be different from all the pi , since p|N but pi N for all i = 1,...,n. So there must be infinitely many primes p ≡ 5(mod6). 1.2 1. Suppose that p|x2 +1. If x is even, clearly p is odd. Then x2 ≡−1(modp) (and as p = 2, we have x2 ≡ 1(modp)), and so x4 ≡ 1(modp).
    [Show full text]
  • Math 331-2 Lecture Notes
    Math 331-2: Abstract Algebra Northwestern University, Lecture Notes Written by Santiago Ca~nez These are notes which provide a basic summary of each lecture for Math 331-2, the second quarter of \MENU: Abstract Algebra", taught by the author at Northwestern University. The book used as a reference is the 3rd edition of Abstract Algebra by Dummit and Foote. Watch out for typos! Comments and suggestions are welcome. Contents Lecture 1: Introduction to Rings 2 Lecture 2: Matrix and Polynomial Rings 6 Lecture 3: More Ring Constructions 8 Lecture 4: Quotient Rings 14 Lecture 5: More on Quotients and Ideals 18 Lecture 6: Maximal Ideals 21 Lecture 7: Prime Ideals 25 Lecture 8: Geometry via Algebra 28 Lecture 9: Rings of Fractions 32 Lecture 10: More on Fractions 37 Lecture 11: Chinese Remainder Theorem 43 Lecture 12: Euclidean Domains 47 Lecture 13: More on Euclidean 51 Lecture 14: Principal Ideal Domains 54 Lecture 15: Unique Factorization 57 Lecture 16: More on UFDs 60 Lecture 17: Polynomials Revisited 63 Lecture 18: Gauss's Lemma 66 Lecture 19: Eisenstein's Criterion 69 Lecture 20: Modules over Rings 72 Lecture 21: Module Homomorphisms 77 Lecture 22: Sums and Free Modules 80 Lecture 23: Modules over PIDs 85 Lecture 24: Structure Theorems 89 Lecture 25: More on Structure Theorems 92 Lecture 26: Jordan Normal Form 97 Lecture 27: More on Normal Forms 103 Lecture 1: Introduction to Rings We continue our study of abstract algebra, this quarter focusing on rings and (to a lesser extent) modules. Rings provide a general setting in which \arithmetic" and the notion of \number" makes sense, and indeed the development of ring theory was borne out of attempts to understand general properties analogous to those of integers.
    [Show full text]
  • Dedekind's 1871 Version of the Theory of Ideals∗
    Dedekind's 1871 version of the theory of ideals¤ Translated by Jeremy Avigad March 19, 2004 Translator's introduction By the middle of the nineteenth century, it had become clear to mathe- maticians that the study of ¯nite ¯eld extensions of the rational numbers is indispensable to number theory, even if one's ultimate goal is to understand properties of diophantine expressions and equations in the ordinary integers. It can happen, however, that the \integers" in such extensions fail to satisfy unique factorization, a property that is central to reasoning about the or- dinary integers. In 1844, Ernst Kummer observed that unique factorization fails for the cyclotomic integers with exponent 23, i.e. the ring Z[³] of inte- gers of the ¯eld Q(³), where ³ is a primitive twenty-third root of unity. In 1847, he published his theory of \ideal divisors" for cyclotomic integers with prime exponent. This was to remedy the situation by introducing, for each such ring of integers, an enlarged domain of divisors, and showing that each integer factors uniquely as a product of these. He did not actually construct these integers, but, rather, showed how one could characterize their behav- ior qua divisibility in terms of ordinary operations on the associated ring of integers. Richard Dedekind and Leopold Kronecker later took up the task of ex- tending the theory to the integers in arbitrary ¯nite extensions of the ratio- nals. Despite their common influences and goals, however, the theories they ¤Work on this translation has been supported by a New Directions fellowship from the Andrew W.
    [Show full text]
  • A History of Stickelberger's Theorem
    A History of Stickelberger’s Theorem A Senior Honors Thesis Presented in Partial Fulfillment of the Requirements for graduation with research distinction in Mathematics in the undergraduate colleges of The Ohio State University by Robert Denomme The Ohio State University June 8, 2009 Project Advisor: Professor Warren Sinnott, Department of Mathematics 1 Contents Introduction 2 Acknowledgements 4 1. Gauss’s Cyclotomy and Quadratic Reciprocity 4 1.1. Solution of the General Equation 4 1.2. Proof of Quadratic Reciprocity 8 2. Jacobi’s Congruence and Cubic Reciprocity 11 2.1. Jacobi Sums 11 2.2. Proof of Cubic Reciprocity 16 3. Kummer’s Unique Factorization and Eisenstein Reciprocity 19 3.1. Ideal Numbers 19 3.2. Proof of Eisenstein Reciprocity 24 4. Stickelberger’s Theorem on Ideal Class Annihilators 28 4.1. Stickelberger’s Theorem 28 5. Iwasawa’s Theory and The Brumer-Stark Conjecture 39 5.1. The Stickelberger Ideal 39 5.2. Catalan’s Conjecture 40 5.3. Brumer-Stark Conjecture 41 6. Conclusions 42 References 42 2 Introduction The late Professor Arnold Ross was well known for his challenge to young students, “Think deeply of simple things.” This attitude applies to no story better than the one on which we are about to embark. This is the century long story of the generalizations of a single idea which first occurred to the 19 year old prodigy, Gauss, and which he was able to write down in no less than 4 pages. The questions that the young genius raised by offering the idea in those 4 pages, however, would torment the greatest minds in all the of the 19th century.
    [Show full text]
  • 12 Prime Ideals
    12 Prime ideals Again, the presentation here is somewhat different than the text. In particular, the sections do not match up. (In fact we have done some of the Chapter 12 material in the text in our Chapter 11 notes.) Our goal is to prove that OK has unique factorization into prime ideals for any number field K. We will use this to determine which primes are of the form x2 + 5y2. 12.1 The ideal factorization theorems We are still missing one main ingredient to prove the existence of factorization into prime ideals. The ingredient we need is the following result. Proposition 12.1. Let K be a number field and I; J be nonzero proper ideals of OK . If J jI and 0 0 I 6= J , then I = JJ for some nonzero proper ideal J of OK . In other words, this notion of J jI for ideals, means I really factors as a product I = JJ 0. This is an immediate consequence of Corollary 12.6 below. Stillwell gives what is perhaps a simpler argument for this in Section 12.5 when K is an imaginary quadratic field. For the moment, let’s take it for granted and see how it implies existence of prime factorization for ideals. Theorem 12.2. (Existence of factorization into prime ideals) Let K be a number field and I be a nonzero proper ideal of OK . Then I = p1 ··· pk where pi’s are prime ideals of OK . Proof. Either I is maximal (and therefore prime) or not. If so we’re done, so suppose not.
    [Show full text]
  • Arxiv:1912.02555V1 [Math.NT]
    DYNAMICAL IDEALS OF NON-COMMUTATIVE RINGS IGOR NIKOLAEV1 Abstract. An analog of the prime ideals for simple non-commutative rings is introduced. We prove the fundamental theorem of arithmetic for such rings. The result is used to classify the surface knots and links in the smooth 4- dimensional manifolds. 1. Introduction The concept of an ideal is fundamental in commutative algebra. Recall that the prime factorization in the ring of integers OK of a number field K fails to be unique. To fix the problem, one needs to complete the set of the prime numbers of OK by the “ideal numbers” lying in an abelian extension of K [Kummer 1847] [4]. A description of the “ideal number” in terms of the ring OK leads to the notion of an ideal [Dedekind 1863] [1]. Formally, the ideal of a commutative ring R is defined as a subset I ⊆ R, such that I is additively closed and IR ⊆ I. However, the true power of the ideals comes from their geometry. For instance, if R is the coordinate ring of an affine variety V , then the prime ideals of R make up a topological space homeomorphic to V . This link between algebra and geometry is critical, e.g. for the Weil Conjectures [Weil 1949] [12]. Although the notion of an ideal adapts to the non-commutative rings using the left, right and two-sided ideals, such an approach seems devoid of meaningful geom- etry. The drawback is that the “coordinate rings” in non-commutative geometry are usually simple, i.e. have only trivial ideals [6, Section 5.3.1].
    [Show full text]
  • Algebraic Number Theory
    Algebraic Number Theory Translated from the German by Norbert Schappacher With 16 Figures Jiirgen Nenkircht Trandator: Norbert Schappacher U.F.R. de Mathkmatique et dkformatique Universitt? Louis Pasteur 7, rue Renk Descartes F-67084 Strasbourg, France e-mail: [email protected] The original German edition was published in 1992 under the title Algebraidze Zahlentheorie ISBN 3-540-54273-6 Springer-Verlag Berlin Heidelberg New York Library of Congress Cataloging-in-Publication Data Neukirch, Jiirgen, 1937-. [Algebraische zahlentheorie. English] Algebraic number theory ! Jiirgen Neukirch, translated from the German by Norbert Schappacher. p. cm. - (Grundlehren der mathematischen Wissenschaften; 322 Inciudes bibliographicai references and index. ISBN 3-540-65399-6 (hc.: alk. paper) 1. Algebraic number theory, I. Title II. Series. QAz47.N51713 1999 512’.74-dcz1 gg-21030 CIP Mathematics Subject Classification (1991): 11-Xx, q-XX ISSN 0072-7830 ISBN 3-540-65399-6 Springer-Veriag Berlin Heidelberg NewYork This work is subject to copyright. All rights are reserved, whether tile whole or part ofthe material is concerned, specifically the rights oftranslation, reprinting, reuse of iliastrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 0 Springer-Verlag Berlin Heidelberg 1999 Printed in Germany Cover design: MetaDesign plus GmbH, Berlin Photocomposed from the translator’s IJTPX files after editing and reformatting by Raymond Seroul, Strasbourg SPIN: 10425040 41/3143-5 4 3 2 1 o Printed on acid-free paper It is a very sad moment for me to write this “Geleitwort” to the English translation of Jtirgen Neukireh’s book on Algebraic Number Theory.
    [Show full text]
  • Fermafs Last Theorem, a Genetic Introduction to Algebraic Number Theory, by Harold M
    218 BOOK REVIEWS REFERENCES 1. P. Dembowski, Finite geometries, Springer-Verlag, Berlin and New York, 1968. 2. D. Foulser, Replaceable translation nets, Proc. London Math. Soc. 22 (1971), 235-264. 3. C. Hering, A new class of quasifields, Math. Z. 118 (1970), 56-57. 4. , On shears of translation planes, Abh. Math. Sem. Ham. 37 (1972), 258-268. 5. D. R. Hughes and F. C. Piper, Projective planes, Springer-Verlag, Berlin and New York, 1973. 6. M. J. Kallaher, On finite affine planes of rank 3, J. Algebra 13 (1969), 544-553. 7. , A survey of weak rank 2 affine planes, Proc. Internat. Conf. on Projecture Planes, (M. J. Kallaher and T. G. Ostrom (eds.)), Wash. St. Univ. Press, Pullman, Washington, 1973. 8. M. J. Kallaher and T. G. Ostrom, Collineation groups irreducible on the components of a translation plane, Geom. Dedicata 9 (1980), 153-194. 9. R. A. Liebler, Finite affine planes of rank three are translation planes, Math. Z. 116 (1970), 89-93. 10. , On finite line transitive affine planes of rank 3, Proc. Internat. Conf. on projective Planes, Wash. St. Univ. Press, Pullman, Washington, 1973. 11. T. G. Ostrom, Linear transformations and collineations of translation planes, J. Algebra 14 (1970), 405-416. 12. , Finite translation planes, Lecture Notes in Math., vol. 158, Springer-Verlag, Berlin and New York, 1970. 13. , Classification of finite translation planes, Proc. Internat. Conf. on Projective Planes, Wash. St. Univ. Press, Pullman, Washington, 1970. 14. , Recent advances in finite translation planes, Foundations of Geometry, (P. Scherk (ed.)), Univ. Toronto Press, Toronto, Canada, 1976.
    [Show full text]
  • Kummer's Theory on Ideal Numbers and Fermat's Last
    KUMMER'S THEORY ON IDEAL NUMBERS AND FERMAT'S LAST THEOREM FIZAY-NOAH LEE Abstract. This paper is an exposition on Ernst Kummer's theory of ideal numbers, which \saves" unique factorization in the ring of integers of the cy- clotomic field. A significant application of this theory is in proving a large subcase of Fermat's Last Theorem, namely, the case of regular prime expo- nents. Contents 1. Cyclotomic Integers 1 2. Cyclotomic Periods 2 3. Divisibility by Prime Divisors 4 4. Ideal Numbers and Divisors 7 5. Class Number and Regular Primes 12 6. Fermat's Last Theorem for Regular Primes 13 7. Acknowledgments 18 References 18 1. Cyclotomic Integers The idea of unique factorization is a familiar concept because it is a property that holds true for the integers and because many properties of the integers depend on it. The failure of unique factorization in the ring of integers of certain cyclotomic fields is what motivated Ernst Kummer to develop his theory of ideal numbers, which restores unique factorization for the rings in question. To begin a study of this theory, we start by investigating the elements of the rings of integers of cyclotomic fields, which Kummer called cyclotomic integers. Definition 1.1. Given a prime integer λ, a primitive λth root of unity is a complex number α that satisfies αλ = 1 and αi 6= 1 for i = 1; 2; :::; λ − 1. 2π 2π Remark 1.2. Concretely, from Euler's formula, one choice of α is cos λ + i sin λ . Observe that α = α−1.
    [Show full text]