Landmark Writings in Western Mathematics 1640-1940
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
QUARTIC CM FIELDS 1. Background the Study of Complex Multiplication
QUARTIC CM FIELDS WENHAN WANG Abstract. In the article, we describe the basic properties, general and specific properties of CM degree 4 fields, as well as illustrating their connection to the study of genus 2 curves with CM. 1. Background The study of complex multiplication is closely related to the study of curves over finite fields and their Jacobian. Basically speaking, for the case of non-supersingular elliptic curves over finite fields, the endomorphism ring is ring-isomorphic to an order in an imaginary quadratic extension K of Q. The structure of imaginary extensions of Q has beenp thoroughly studied, and the ringsq of integers are simply generated by f1; Dg if D ≡ 1 mod 4, f D g ≡ or by 1; 4 if D 0 mod 4, where D is the discriminant of the field K. The theory of complex multiplication can be carried from elliptic curves to the (Jacobians) of genus 2 (hyperelliptic) curves. More explicitly, the Jacobian of any non-supersingular genus 2 (and hence, hyperelliptic) curve defined over a finite field has CM by an order in a degree 4, or quartic extension over Q, where the extension field K has to be totally imaginary. Description of the endomorphism ring of the Jacobian of a genus 2 curve over a finite field largely depends on the field K for which the curve has CM by. Many articles in the area of the study of genus two curves lead to the study of many properties of the field K. Hence the main goal of this article is, based on the knowledge of the author in the study of the genus 2 curves over finite fields, to give a survey of various, general or specific, properties of degree 4 CM fields. -
Of the American Mathematical Society August 2017 Volume 64, Number 7
ISSN 0002-9920 (print) ISSN 1088-9477 (online) of the American Mathematical Society August 2017 Volume 64, Number 7 The Mathematics of Gravitational Waves: A Two-Part Feature page 684 The Travel Ban: Affected Mathematicians Tell Their Stories page 678 The Global Math Project: Uplifting Mathematics for All page 712 2015–2016 Doctoral Degrees Conferred page 727 Gravitational waves are produced by black holes spiraling inward (see page 674). American Mathematical Society LEARNING ® MEDIA MATHSCINET ONLINE RESOURCES MATHEMATICS WASHINGTON, DC CONFERENCES MATHEMATICAL INCLUSION REVIEWS STUDENTS MENTORING PROFESSION GRAD PUBLISHING STUDENTS OUTREACH TOOLS EMPLOYMENT MATH VISUALIZATIONS EXCLUSION TEACHING CAREERS MATH STEM ART REVIEWS MEETINGS FUNDING WORKSHOPS BOOKS EDUCATION MATH ADVOCACY NETWORKING DIVERSITY blogs.ams.org Notices of the American Mathematical Society August 2017 FEATURED 684684 718 26 678 Gravitational Waves The Graduate Student The Travel Ban: Affected Introduction Section Mathematicians Tell Their by Christina Sormani Karen E. Smith Interview Stories How the Green Light was Given for by Laure Flapan Gravitational Wave Research by Alexander Diaz-Lopez, Allyn by C. Denson Hill and Paweł Nurowski WHAT IS...a CR Submanifold? Jackson, and Stephen Kennedy by Phillip S. Harrington and Andrew Gravitational Waves and Their Raich Mathematics by Lydia Bieri, David Garfinkle, and Nicolás Yunes This season of the Perseid meteor shower August 12 and the third sighting in June make our cover feature on the discovery of gravitational waves -
Hilbert Constance Reid
Hilbert Constance Reid Hilbert c COPERNICUS AN IMPRINT OF SPRINGER-VERLAG © 1996 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc in 1996 All rights reserved. No part ofthis publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Library ofCongress Cataloging·in-Publication Data Reid, Constance. Hilbert/Constance Reid. p. Ctn. Originally published: Berlin; New York: Springer-Verlag, 1970. Includes bibliographical references and index. ISBN 978-0-387-94674-0 ISBN 978-1-4612-0739-9 (eBook) DOI 10.1007/978-1-4612-0739-9 I. Hilbert, David, 1862-1943. 2. Mathematicians-Germany Biography. 1. Title. QA29.HsR4 1996 SIO'.92-dc20 [B] 96-33753 Manufactured in the United States of America. Printed on acid-free paper. 9 8 7 6 543 2 1 ISBN 978-0-387-94674-0 SPIN 10524543 Questions upon Rereading Hilbert By 1965 I had written several popular books, such as From liro to Infinity and A Long Way from Euclid, in which I had attempted to explain certain easily grasped, although often quite sophisticated, mathematical ideas for readers very much like myself-interested in mathematics but essentially untrained. At this point, with almost no mathematical training and never having done any bio graphical writing, I became determined to write the life of David Hilbert, whom many considered the profoundest mathematician of the early part of the 20th century. Now, thirty years later, rereading Hilbert, certain questions come to my mind. -
Number Theoretic Symbols in K-Theory and Motivic Homotopy Theory
Number Theoretic Symbols in K-theory and Motivic Homotopy Theory Håkon Kolderup Master’s Thesis, Spring 2016 Abstract We start out by reviewing the theory of symbols over number fields, emphasizing how this notion relates to classical reciprocity lawsp and algebraic pK-theory. Then we compute the second algebraic K-group of the fields pQ( −1) and Q( −3) based on Tate’s technique for K2(Q), and relate the result for Q( −1) to the law of biquadratic reciprocity. We then move into the realm of motivic homotopy theory, aiming to explain how symbols in number theory and relations in K-theory and Witt theory can be described as certain operations in stable motivic homotopy theory. We discuss Hu and Kriz’ proof of the fact that the Steinberg relation holds in the ring π∗α1 of stable motivic homotopy groups of the sphere spectrum 1. Based on this result, Morel identified the ring π∗α1 as MW the Milnor-Witt K-theory K∗ (F ) of the ground field F . Our last aim is to compute this ring in a few basic examples. i Contents Introduction iii 1 Results from Algebraic Number Theory 1 1.1 Reciprocity laws . 1 1.2 Preliminary results on quadratic fields . 4 1.3 The Gaussian integers . 6 1.3.1 Local structure . 8 1.4 The Eisenstein integers . 9 1.5 Class field theory . 11 1.5.1 On the higher unit groups . 12 1.5.2 Frobenius . 13 1.5.3 Local and global class field theory . 14 1.6 Symbols over number fields . -
History of Mathematics in Mathematics Education. Recent Developments Kathy Clark, Tinne Kjeldsen, Sebastian Schorcht, Constantinos Tzanakis, Xiaoqin Wang
History of mathematics in mathematics education. Recent developments Kathy Clark, Tinne Kjeldsen, Sebastian Schorcht, Constantinos Tzanakis, Xiaoqin Wang To cite this version: Kathy Clark, Tinne Kjeldsen, Sebastian Schorcht, Constantinos Tzanakis, Xiaoqin Wang. History of mathematics in mathematics education. Recent developments. History and Pedagogy of Mathematics, Jul 2016, Montpellier, France. hal-01349230 HAL Id: hal-01349230 https://hal.archives-ouvertes.fr/hal-01349230 Submitted on 27 Jul 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. HISTORY OF MATHEMATICS IN MATHEMATICS EDUCATION Recent developments Kathleen CLARK, Tinne Hoff KJELDSEN, Sebastian SCHORCHT, Constantinos TZANAKIS, Xiaoqin WANG School of Teacher Education, Florida State University, Tallahassee, FL 32306-4459, USA [email protected] Department of Mathematical Sciences, University of Copenhagen, Denmark [email protected] Justus Liebig University Giessen, Germany [email protected] Department of Education, University of Crete, Rethymnon 74100, Greece [email protected] Department of Mathematics, East China Normal University, China [email protected] ABSTRACT This is a survey on the recent developments (since 2000) concerning research on the relations between History and Pedagogy of Mathematics (the HPM domain). Section 1 explains the rationale of the study and formulates the key issues. -
EVERY ABELIAN GROUP IS the CLASS GROUP of a SIMPLE DEDEKIND DOMAIN 2 Class Group
EVERY ABELIAN GROUP IS THE CLASS GROUP OF A SIMPLE DEDEKIND DOMAIN DANIEL SMERTNIG Abstract. A classical result of Claborn states that every abelian group is the class group of a commutative Dedekind domain. Among noncommutative Dedekind prime rings, apart from PI rings, the simple Dedekind domains form a second important class. We show that every abelian group is the class group of a noncommutative simple Dedekind domain. This solves an open problem stated by Levy and Robson in their recent monograph on hereditary Noetherian prime rings. 1. Introduction Throughout the paper, a domain is a not necessarily commutative unital ring in which the zero element is the unique zero divisor. In [Cla66], Claborn showed that every abelian group G is the class group of a commutative Dedekind domain. An exposition is contained in [Fos73, Chapter III §14]. Similar existence results, yield- ing commutative Dedekind domains which are more geometric, respectively num- ber theoretic, in nature, were obtained by Leedham-Green in [LG72] and Rosen in [Ros73, Ros76]. Recently, Clark in [Cla09] showed that every abelian group is the class group of an elliptic commutative Dedekind domain, and that this domain can be chosen to be the integral closure of a PID in a quadratic field extension. See Clark’s article for an overview of his and earlier results. In commutative mul- tiplicative ideal theory also the distribution of nonzero prime ideals within the ideal classes plays an important role. For an overview of realization results in this direction see [GHK06, Chapter 3.7c]. A ring R is a Dedekind prime ring if every nonzero submodule of a (left or right) progenerator is a progenerator (see [MR01, Chapter 5]). -
Quadratic Forms, Lattices, and Ideal Classes
Quadratic forms, lattices, and ideal classes Katherine E. Stange March 1, 2021 1 Introduction These notes are meant to be a self-contained, modern, simple and concise treat- ment of the very classical correspondence between quadratic forms and ideal classes. In my personal mental landscape, this correspondence is most naturally mediated by the study of complex lattices. I think taking this perspective breaks the equivalence between forms and ideal classes into discrete steps each of which is satisfyingly inevitable. These notes follow no particular treatment from the literature. But it may perhaps be more accurate to say that they follow all of them, because I am repeating a story so well-worn as to be pervasive in modern number theory, and nowdays absorbed osmotically. These notes require a familiarity with the basic number theory of quadratic fields, including the ring of integers, ideal class group, and discriminant. I leave out some details that can easily be verified by the reader. A much fuller treatment can be found in Cox's book Primes of the form x2 + ny2. 2 Moduli of lattices We introduce the upper half plane and show that, under the quotient by a natural SL(2; Z) action, it can be interpreted as the moduli space of complex lattices. The upper half plane is defined as the `upper' half of the complex plane, namely h = fx + iy : y > 0g ⊆ C: If τ 2 h, we interpret it as a complex lattice Λτ := Z+τZ ⊆ C. Two complex lattices Λ and Λ0 are said to be homothetic if one is obtained from the other by scaling by a complex number (geometrically, rotation and dilation). -
The Kronecker-Weber Theorem
The Kronecker-Weber Theorem Lucas Culler Introduction The Kronecker-Weber theorem is one of the earliest known results in class field theory. It says: Theorem. (Kronecker-Weber-Hilbert) Every abelian extension of the rational numbers Q is con- tained in a cyclotomic extension. Recall that an abelian extension is a finite field extension K/Q such that the galois group Gal(K/Q) th is abelian, and a cyclotomic extension is an extension of the form Q(ζ), where ζ is an n root of unity. This paper consists of two proofs of the Kronecker-Weber theorem. The first is rather involved, but elementary, and uses the theory of higher ramification groups. The second is a simple application of the main results of class field theory, which classifies abelian extension of an arbitrary number field. An Elementary Proof Now we will present an elementary proof of the Kronecker-Weber theoerem, in the spirit of Hilbert’s original proof. The particular strategy used here is given as a series of exercises in Marcus [1]. Minkowski’s Theorem We first prove a classical result due to Minkowski. Theorem. (Minkowski) Any finite extension of Q has nonzero discriminant. In particular, such an extension is ramified at some prime p ∈ Z. Proof. Let K/Q be a finite extension of degree n, and let A = OK be its ring of integers. Consider the embedding: r s A −→ R ⊕ C x 7→ (σ1(x), ..., σr(x), τ1(x), ..., τs(x)) where the σi are the real embeddings of K and the τi are the complex embeddings, with one embedding chosen from each conjugate pair, so that n = r + 2s. -
Infinite Galois Theory
Infinite Galois Theory Haoran Liu May 1, 2016 1 Introduction For an finite Galois extension E/F, the fundamental theorem of Galois Theory establishes an one-to-one correspondence between the intermediate fields of E/F and the subgroups of Gal(E/F), the Galois group of the extension. With this correspondence, we can examine the the finite field extension by using the well-established group theory. Naturally, we wonder if this correspondence still holds if the Galois extension E/F is infinite. It is very tempting to assume the one-to-one correspondence still exists. Unfortu- nately, there is not necessary a correspondence between the intermediate fields of E/F and the subgroups of Gal(E/F)whenE/F is a infinite Galois extension. It will be illustrated in the following example. Example 1.1. Let F be Q,andE be the splitting field of a set of polynomials in the form of x2 p, where p is a prime number in Z+. Since each automorphism of E that fixes F − is determined by the square root of a prime, thusAut(E/F)isainfinitedimensionalvector space over F2. Since the number of homomorphisms from Aut(E/F)toF2 is uncountable, which means that there are uncountably many subgroups of Aut(E/F)withindex2.while the number of subfields of E that have degree 2 over F is countable, thus there is no bijection between the set of all subfields of E containing F and the set of all subgroups of Gal(E/F). Since a infinite Galois group Gal(E/F)normally have ”too much” subgroups, there is no subfield of E containing F can correspond to most of its subgroups. -
Local-Global Methods in Algebraic Number Theory
LOCAL-GLOBAL METHODS IN ALGEBRAIC NUMBER THEORY ZACHARY KIRSCHE Abstract. This paper seeks to develop the key ideas behind some local-global methods in algebraic number theory. To this end, we first develop the theory of local fields associated to an algebraic number field. We then describe the Hilbert reciprocity law and show how it can be used to develop a proof of the classical Hasse-Minkowski theorem about quadratic forms over algebraic number fields. We also discuss the ramification theory of places and develop the theory of quaternion algebras to show how local-global methods can also be applied in this case. Contents 1. Local fields 1 1.1. Absolute values and completions 2 1.2. Classifying absolute values 3 1.3. Global fields 4 2. The p-adic numbers 5 2.1. The Chevalley-Warning theorem 5 2.2. The p-adic integers 6 2.3. Hensel's lemma 7 3. The Hasse-Minkowski theorem 8 3.1. The Hilbert symbol 8 3.2. The Hasse-Minkowski theorem 9 3.3. Applications and further results 9 4. Other local-global principles 10 4.1. The ramification theory of places 10 4.2. Quaternion algebras 12 Acknowledgments 13 References 13 1. Local fields In this section, we will develop the theory of local fields. We will first introduce local fields in the special case of algebraic number fields. This special case will be the main focus of the remainder of the paper, though at the end of this section we will include some remarks about more general global fields and connections to algebraic geometry. -
A Second Course in Algebraic Number Theory
A second course in Algebraic Number Theory Vlad Dockchitser Prerequisites: • Galois Theory • Representation Theory Overview: ∗ 1. Number Fields (Review, K; OK ; O ; ClK ; etc) 2. Decomposition of primes (how primes behave in eld extensions and what does Galois's do) 3. L-series (Dirichlet's Theorem on primes in arithmetic progression, Artin L-functions, Cheboterev's density theorem) 1 Number Fields 1.1 Rings of integers Denition 1.1. A number eld is a nite extension of Q Denition 1.2. An algebraic integer α is an algebraic number that satises a monic polynomial with integer coecients Denition 1.3. Let K be a number eld. It's ring of integer OK consists of the elements of K which are algebraic integers Proposition 1.4. 1. OK is a (Noetherian) Ring 2. , i.e., ∼ [K:Q] as an abelian group rkZ OK = [K : Q] OK = Z 3. Each can be written as with and α 2 K α = β=n β 2 OK n 2 Z Example. K OK Q Z ( p p [ a] a ≡ 2; 3 mod 4 ( , square free) Z p Q( a) a 2 Z n f0; 1g a 1+ a Z[ 2 ] a ≡ 1 mod 4 where is a primitive th root of unity Q(ζn) ζn n Z[ζn] Proposition 1.5. 1. OK is the maximal subring of K which is nitely generated as an abelian group 2. O`K is integrally closed - if f 2 OK [x] is monic and f(α) = 0 for some α 2 K, then α 2 OK . Example (Of Factorisation). -
Early History of the Riemann Hypothesis in Positive Characteristic
The Legacy of Bernhard Riemann c Higher Education Press After One Hundred and Fifty Years and International Press ALM 35, pp. 595–631 Beijing–Boston Early History of the Riemann Hypothesis in Positive Characteristic Frans Oort∗ , Norbert Schappacher† Abstract The classical Riemann Hypothesis RH is among the most prominent unsolved prob- lems in modern mathematics. The development of Number Theory in the 19th century spawned an arithmetic theory of polynomials over finite fields in which an analogue of the Riemann Hypothesis suggested itself. We describe the history of this topic essentially between 1920 and 1940. This includes the proof of the ana- logue of the Riemann Hyothesis for elliptic curves over a finite field, and various ideas about how to generalize this to curves of higher genus. The 1930ies were also a period of conflicting views about the right method to approach this problem. The later history, from the proof by Weil of the Riemann Hypothesis in charac- teristic p for all algebraic curves over a finite field, to the Weil conjectures, proofs by Grothendieck, Deligne and many others, as well as developments up to now are described in the second part of this diptych: [44]. 2000 Mathematics Subject Classification: 14G15, 11M99, 14H52. Keywords and Phrases: Riemann Hypothesis, rational points over a finite field. Contents 1 From Richard Dedekind to Emil Artin 597 2 Some formulas for zeta functions. The Riemann Hypothesis in characteristic p 600 3 F.K. Schmidt 603 ∗Department of Mathematics, Utrecht University, Princetonplein 5, 3584 CC