DOCSLIB.ORG

View This Volume's Front and Back Matter

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
View This Volume's Front and Back Matter http://dx.doi.org/10.1090/fim/021 FIELDS INSTITUT E MONOGRAPHS THE FIELD S INSTITUT E FO R RESEARC H I N MATHEMATICA L SCIENCE S Brauer Typ e Embedding Problems Arne Lede t American Mathematica l Societ y Providence, Rhode Islan d The Field s Institut e for Researc h i n Mathematical Science s The Field s Institute i s a center fo r mathematica l researc h activity , locate d i n Toronto , Canada. Ou r missio n i s to provid e a supportiv e an d stimulatin g environmen t fo r math - ematics research , innovatio n an d education . Th e Institut e i s supporte d b y th e Ontari o Ministry o f Training, College s and Universities , the Natural Science s and Engineerin g Re - search Counci l o f Canada , an d seve n Ontari o universitie s (Carleton , McMaster , Ottawa , Toronto, Waterloo , Wester n Ontario , an d York) . I n additio n ther e ar e severa l affiliate d universities an d corporat e sponsor s i n both Canad a an d th e Unite d States . Fields Institut e Editoria l Board : Car l R . Rieh m (Managin g Editor) , Barbar a Le e Keyfitz (Directo r o f the Institute) , Thoma s S . Salisbur y (Deput y Director) , Joh n Blan d (Toronto), Kennet h R . Davidso n (Waterloo) , Joe l Feldma n (UBC) , R . Mar k Goresk y (Institute fo r Advance d Study , Princeton) , Camero n Stewar t (Waterloo) , Norik o Yu i (Queen's). The autho r wa s supporte d i n par t b y a Queen' s Universit y Advisor y Researc h Com - mittee Postdoctora l Fellowship . 2000 Mathematics Subject Classification. Primar y 12F12 , 16K50 ; Secondar y 16S35 , 12G05. For additiona l informatio n an d update s o n this book , visi t www.ams.org/bookpages/fim-21 Library o f Congres s Cataloging-in-Publicatio n Dat a Ledet, Arne , 1967 - Brauer typ e embeddin g problem s / Arn e Ledet . p. cm . (Field s Institut e monographs , ISS N 1069-5273 ; 21 ) Includes bibliographica l reference s an d index . ISBN 0-8218-3726- 5 (alk . paper ) 1. Inverse Galoi s theory . 2 . Brauer groups . I . Title . II . Series . QA247 + 200504281 3 Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitted t o mak e fai r us e o f the material , suc h a s to cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f the sourc e i s given . Republication, systemati c copying , o r multiple reproduction o f any material i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addressed t o the Acquisition s Department , America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s ca n als o b e mad e b y e-mail t o [email protected] . © 200 5 b y the America n Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . This publicatio n wa s prepare d b y The Field s Institute . Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 1 0 09 0 8 0 7 0 6 0 5 Contents Introduction v Acknowledgments vii i Chapter 1 . Galoi s Theor y 1 Notation 1 1.1. Integra l element s 1 1.2. Algebrai c extension s an d splittin g fields 2 1.3. Separabilit y 4 1.4. Galoi s extensions 6 1.5. Th e Galoi s group a s permutation grou p 9 1.6. Reductio n modul o prime s 1 1 1.7. Cyclotomi c fields 1 3 1.8. Cycli c extensions an d Hilber t Sat z 9 0 1 5 1.9. Radica l extension s an d solvabilit y 1 7 1.10. Th e norma l basi s theore m 1 9 Exercises 2 1 Chapter 2 . Invers e Galoi s Theor y an d Embeddin g Problem s 2 7 2.1. Invers e Galoi s theory 2 7 2.2. Definitio n o f Galoi s theoretical embeddin g problem s 2 9 2.3. Grou p cohomolog y 3 0 2.4. Braue r typ e embeddin g problem s 3 4 Exercises 4 0 Chapter 3 . Braue r Group s 4 3 3.1. Basi c fact s abou t algebra s 4 3 3.2. Simpl e Artinia n ring s 4 4 3.3. Finite-dimensiona l centra l simpl e algebra s 4 6 3.4. Th e Braue r grou p 5 0 3.5. Cycli c algebras 5 3 3.6. Braue r typ e embeddin g problem s revisite d 5 6 3.7. Extension s o f the dihedra l grou p 5 9 Exercises 6 2 Chapter 4 . Grou p Cohomolog y 6 7 4.1. Th e cohomologie s i n dimension s 0 and 1 6 7 4.2. Galoi s Twis t 6 8 4.3. Th e long-exac t cohomolog y sequenc e 7 0 4.4. Th e corestrictio n ma p 7 2 iv Content s Exercises 7 3 Chapter 5 . Quadrati c Form s 7 7 5.1. Basi c theory o f quadratic form s 7 7 5.2. Cliffor d algebra s 8 1 5.3. Quadrati c form s an d non-abelia n cohomolog y 8 3 Exercises 8 5 Chapter 6 . Decomposin g th e Obstructio n 8 7 6.1. Decompositio n alon g a direct produc t 8 7 6.2. Orthogona l representation s 9 5 6.3. Projectiv e representation s 9 8 Exercises 10 2 Chapter 7 . Quadrati c Form s an d Embeddin g Problem s 10 5 7.1. Group s o f exponent 4 10 5 7.2. Witt' s Criterio n 10 8 7.3. Group s o f exponent 8 11 4 Exercises 12 1 Chapter 8 . Reducin g th e Embeddin g Proble m 12 3 8.1. Reductio n t o Brauer typ e embeddin g problem s 12 3 8.2. Examples : Th e holomorp h o f the quaternio n grou p 13 3 8.3. Embeddin g problem s wit h cycli c kernel o f order 4 14 1 Exercises 14 8 Appendix A . Pro-finit e Galoi s Theor y 15 1 Introduction 15 1 A.l. Th e separabl e closur e o f a field 15 1 A.2. Galoi s extension s 15 2 A.3. Pro-finit e group s 15 4 A.4. Pro-finit e Galoi s theory 15 6 A.5. Pro-finit e grou p cohomolog y 15 7 A.6. Pro-finit e cohomolog y an d th e Braue r grou p 15 8 Exercises 16 0 Bibliography 16 3 Index 167 Introduction Consider a Galoi s theoretica l embeddin g problem , i.e. , th e questio n o f embeddin g a Galoi s extensio n M/K wit h Galoi s grou p G — Gal(M/K) int o a large r Galoi s extension F/K, suc h tha t th e Galoi s grou p Gal(F/K) i s isomorphi c t o a specifie d group E an d th e restrictio n ma p fro m Ga l (F/if) t o G correspond s t o a give n homomorphism n: E — > G. Ho w d o w e approac h suc h a problem ? Ho w d o w e determine whethe r suc h a n extensio n F/K exists ? Ho w d o w e find i t i f it does ? O r (preferably) ho w d o w e find al l o f them i f there ar e any ? The answer s t o thes e question s o f cours e depen d o n th e kin d o f embeddin g problem considered , bot h with respec t t o the nature o f the groups an d the nature o f the field. Fo r instance, i f G and E ar e cyclic , it takes nothing mor e than elementar y Galois theor y t o solv e th e proble m ove r a finite field, wherea s i t take s clas s field theory (o r a t leas t som e reasonably sophisticate d algebrai c numbe r theory ) t o solv e it ove r th e field o f rational numbers .
Load more

Recommended publications
  • Phase Transition on the Toeplitz Algebra of the Affine Semigroup Over the Natural Numbers
    University of Wollongong Research Online Faculty of Engineering and Information Faculty of Informatics - Papers (Archive) Sciences 1-1-2010 Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers Marcelo Laca University of Newcastle, Australia Iain F. Raeburn University of Wollongong, [email protected] Follow this and additional works at: https://ro.uow.edu.au/infopapers Part of the Physical Sciences and Mathematics Commons Recommended Citation Laca, Marcelo and Raeburn, Iain F.: Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers 2010, 643-688. https://ro.uow.edu.au/infopapers/1749 Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: [email protected] Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers Abstract We show that the group of orientation-preserving affine ansformationstr of the rational numbers is quasi- lattice ordered by its subsemigroup N⋊N×. The associated Toeplitz C∗-algebra T(N⋊N×) is universal for isometric representations which are covariant in the sense of Nica. We give a presentation of T(N⋊N×) in terms of generators and relations, and use this to show that the C∗-algebra QN recently introduced by Cuntz is the boundary quotient of in the sense of Crisp and Laca. The Toeplitz algebra T(N⋊N×) carries a natural dynamics σ, which induces the one considered by Cuntz on the quotient QN, and our main result is the computation of the KMSβ (equilibrium) states of the dynamical system (T(N⋊N×),R,σ) for all values of the inverse temperature β.
    [Show full text]
  • Monoid Kernels and Profinite Topologies on the Free Abelian Group
    BULL. AUSTRAL. MATH. SOC. 20M35, 20E18, 20M07, 20K45 VOL. 60 (1999) [391-402] MONOID KERNELS AND PROFINITE TOPOLOGIES ON THE FREE ABELIAN GROUP BENJAMIN STEINBERG To each pseudovariety of Abelian groups residually containing the integers, there is naturally associated a profinite topology on any finite rank free Abelian group. We show in this paper that if the pseudovariety in question has a decidable membership problem, then one can effectively compute membership in the closure of a subgroup and, more generally, in the closure of a rational subset of such a free Abelian group. Several applications to monoid kernels and finite monoid theory are discussed. 1. INTRODUCTION In the early 1990's, the Rhodes' type II conjecture was positively answered by Ash [2] and independently by Ribes and Zalesskii [10]. The type II submonoid, or G-kernel, of a finite monoid is the set of all elements which relate to 1 under any relational morphism with a finite group. Equivalently, an element is of type II if 1 is in the closure of a certain rational language associated to that element in the profinite topology on a free group. The approach of Ribes and Zalesskii was based on calculating the profinite closure of a rational subset of a free group. They were also able to use this approach to calculate the Gp-kernel of a finite monoid. In both cases, it turns out the the closure of a rational subset of the free group in the appropriate profinite topology is again rational. See [7] for a survey of the type II theorem and its motivation.
    [Show full text]
  • (I) Eu ¿ 0 ; Received by the Editors November 23, 1992 And, in Revised Form, April 19, 1993
    proceedings of the american mathematical society Volume 123, Number 1, January 1995 TRIANGULAR UHF ALGEBRAS OVER ARBITRARYFIELDS R. L. BAKER (Communicated by Palle E. T. Jorgensen) Abstract. Let K be an arbitrary field. Let (qn) be a sequence of posi- tive integers, and let there be given a family \f¥nm\n > m} of unital K- monomorphisms *F„m. Tqm(K) —►Tq„(K) such that *¥np*¥pm= %im when- ever m < n , where Tq„(K) is the if-algebra of all q„ x q„ upper trian- gular matrices over K. A triangular UHF (TUHF) K-algebra is any K- algebra that is A'-isomorphic to an algebraic inductive limit of the form &~ = lim( Tq„(K) ; W„m). The first result of the paper is that if the embeddings ^Vnm satisfy certain natural dimensionality conditions and if £? = lim (TPn(K) ; Onm) is an arbitrary TUHF if-algebra, then 5? is A-isomorphic to ^ , only if the supernatural number, N[{pn)], of (q„) is less than or equal to the super- natural number, N[(p„)], of (pn). Thus, if the embeddings i>„m also satisfy the above dimensionality conditions, then S? is À"-isomorphic to F , only if NliPn)] = N[(qn)] ■ The second result of the paper is a nontrivial "triangular" version of the fact that if p, q are positive integers, then there exists a unital A"-monomorphism <P: Mq{K) —>MP(K), only if q\p . The first result of the paper depends directly on the second result. 1. Introduction Let K be a fixed but arbitrary field. In this paper we investigate a certain class of A.-algebras (i.e., algebras over K) called triangular UHF (TUHF) K- algebras.
    [Show full text]
  • Profinite Surface Groups and the Congruence Kernel Of
    Pro¯nite surface groups and the congruence kernel of arithmetic lattices in SL2(R) P. A. Zalesskii* Abstract Let X be a proper, nonsingular, connected algebraic curve of genus g over the ¯eld C of complex numbers. The algebraic fundamental group ¡ = ¼1(X) in the sense of SGA-1 [1971] is the pro¯nite completion of top the fundamental group ¼1 (X) of a compact oriented 2-manifold. We prove that every projective normal (respectively, caracteristic, accessible) subgroup of ¡ is isomorphic to a normal (respectively, caracteristic, accessible) subgroup of a free pro¯nite group. We use this description to give a complete solution of the congruence subgroup problem for aritmetic lattices in SL2(R). Introduction Let X be a proper, nonsingular, connected algebraic curve of genus g over the ¯eld C of complex numbers. As a topological space X is a compact oriented 2-manifold and is simply a sphere with g handles top added. The group ¦ = ¼1 (X) is called a surface group and has 2g generators ai; bi (i = 1; : : : ; g) subject to one relation [a1; b1][a2; b2] ¢ ¢ ¢ [ag; bg] = 1. The subgroup structure of ¦ is well known. Namely, if H is a subgroup of ¦ of ¯nite index n, then H is again a surface group of genus n(g ¡ 1) + 1. If n is in¯nite, then H is free. The algebraic fundamental group ¡ = ¼1(X) in the sense of SGA-1 [1971] is the pro¯nite completion top of the fundamental group ¼1 (X) in the topological sense (see Exp. 10, p. 272 in [SGA-1]).
    [Show full text]
  • One Dimensional Groups Definable in the P-Adic Numbers
    One dimensional groups definable in the p-adic numbers Juan Pablo Acosta L´opez November 27, 2018 Abstract A complete list of one dimensional groups definable in the p-adic num- bers is given, up to a finite index subgroup and a quotient by a finite subgroup. 1 Introduction The main result of the following is Proposition 8.1. There we give a complete list of all one dimensional groups definable in the p-adics, up to finite index and quotient by finite kernel. This is similar to the list of all groups definable in a real closed field which are connected and one dimensional obtained in [7]. The starting point is the following result. Proposition 1.1. Suppose T is a complete first order theory in a language extending the language of rings, such that T extends the theory of fields. Suppose T is algebraically bounded and dependent. Let G be a definable group. If G arXiv:1811.09854v1 [math.LO] 24 Nov 2018 is definably amenable then there is an algebraic group H and a type-definable subgroup T G of bounded index and a type-definable group morphism T H with finite kernel.⊂ → This is found in Theorem 2.19 of [8]. We just note that algebraic boundedness is seen in greater generality than needed in [5], and it can also be extracted from the proofs in cell decomposition [4]. That one dimensional groups are definably amenable comes from the following result. 1 Proposition 1.2. Suppose G is a definable group in Qp which is one dimen- sional.
    [Show full text]
  • MATHEMATICS Algebra, Geometry, Combinatorics
    MATHEMATICS Algebra, geometry, combinatorics Dr Mark V Lawson November 17, 2014 ii Contents Preface v Introduction vii 1 The nature of mathematics 1 1.1 The scope of mathematics . .1 1.2 Pure versus applied mathematics . .3 1.3 The antiquity of mathematics . .4 1.4 The modernity of mathematics . .6 1.5 The legacy of the Greeks . .8 1.6 The legacy of the Romans . .8 1.7 What they didn't tell you in school . .9 1.8 Further reading and links . 10 2 Proofs 13 2.1 How do we know what we think is true is true? . 14 2.2 Three fundamental assumptions of logic . 16 2.3 Examples of proofs . 17 2.3.1 Proof 1 . 17 2.3.2 Proof 2 . 20 2.3.3 Proof 3 . 22 2.3.4 Proof 4 . 23 2.3.5 Proof 5 . 25 2.4 Axioms . 31 2.5 Mathematics and the real world . 35 2.6 Proving something false . 35 2.7 Key points . 36 2.8 Mathematical creativity . 37 i ii CONTENTS 2.9 Set theory: the language of mathematics . 37 2.10 Proof by induction . 46 3 High-school algebra revisited 51 3.1 The rules of the game . 51 3.1.1 The axioms . 51 3.1.2 Indices . 57 3.1.3 Sigma notation . 60 3.1.4 Infinite sums . 62 3.2 Solving quadratic equations . 64 3.3 *Order . 70 3.4 *The real numbers . 71 4 Number theory 75 4.1 The remainder theorem . 75 4.2 Greatest common divisors .
    [Show full text]
  • Valuations, Orderings, and Milnor K-Theory Mathematical Surveys and Monographs
    http://dx.doi.org/10.1090/surv/124 Valuations, Orderings, and Milnor K-Theory Mathematical Surveys and Monographs Volume 124 Valuations, Orderings, and Milnor K-Theory Ido Efrat American Mathematical Society EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 12J10, 12J15; Secondary 12E30, 12J20, 19F99. For additional information and updates on this book, visit www.ams.org/bookpages/surv-124 Library of Congress Cataloging-in-Publication Data Efrat, Ido, 1963- Valuations, orderings, and Milnor if-theory / Ido Efrat. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 124) Includes bibliographical references and index. ISBN 0-8218-4041-X (alk. paper) 1. Valuation theory. 2. Ordered fields. 3. X-theory I. Title. II. Mathematical surveys and monographs ; no. 124. QA247.E3835 2006 515/.78-dc22 2005057091 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2006 by the American Mathematical Society.
    [Show full text]
  • P-Adic L-Functions and Modular Forms
    p-adic L-functions and modular forms Alexei PANCHISHKIN Institut Fourier, Université Grenoble-1 B.P.74, 38402 St.Martin d'Hères, FRANCE Lectures for the International Summer School and Conference p-adic Analysis and its Applications August 31th to September 18th, 2009 (Trieste - ITALY) September, 2009 Alexei PANCHISHKIN (Grenoble) p-adic L-functions and modular forms ICTP, September,2009 1 / 1 Abstract 1) Congruences and p-adic numbers. Hensel's lemma. The Tate eld 2) Continuous and p-adic analytic functions. Mahler's criterion. Newton polygons Zeroes of analytic functions. The Weierstrass preparation theorem and its generalizations. 3) Distributions, measures, and the abstract Kummer congruences. The Kubota and Leopoldt p-adic L-functions and the Iwasawa algebra 4) Modular forms and L-functions. Congruences and families of modular forms. 5) Method of canonical projection of modular distributions. Examples of construction of p-adic L-functions Alexei PANCHISHKIN (Grenoble) p-adic L-functions and modular forms ICTP, September,2009 2 / 1 Related topics (for discussions, not included in the text of these materials) 6) Other approaches of constructing p-adic L-functions by Mazur-Tate-Teitelbaum, J.Coates, P.Colmez, H.Hida ... (using modular symbols by Manin-Mazur and their generalizations; Euler systems, work of D.Delbourgo, T.Ochiai, L.Berger, ..., overconvergent modular symbols by R.Pollack, G.Stevens, H.Darmon, ...) 7) Relations to the Iwasawa Theory 8) Applications of p-adic L-functions to Diophantine geometry 9) Open questions and problems in the theory of p-adic L-functions (Basic sources: Coates 60th Birthday Volume, Bourbaki talks by P.Colmez, J.Coates ...) Alexei PANCHISHKIN (Grenoble) p-adic L-functions and modular forms ICTP, September,2009 3 / 1 Lecture N◦1.
    [Show full text]
  • A Finiteness Theorem for Odd Perfect Numbers
    A finiteness theorem for odd perfect numbers Paul Pollack University of Georgia June 23, 2020 1 of 23 Ex: 5 is deficient (s(5) = 1), 12 is abundant (s(12) = 16), and 6 is perfect (s(6) = 6). Three kinds of natural numbers Among simple even numbers, some are superabundant, others are deficient: these two classes are as two extremes opposed one to the other; as for those that occupy the middle point between the two, they are said to be perfect. { Nicomachus (ca. 100 AD), Introductio Arithmetica P Let s(n) = djn;d<n d be the sum of the proper divisors of n. Abundant: s(n) > n. Deficient: s(n) < n. Perfect: s(n) = n. 2 of 23 Three kinds of natural numbers Among simple even numbers, some are superabundant, others are deficient: these two classes are as two extremes opposed one to the other; as for those that occupy the middle point between the two, they are said to be perfect. { Nicomachus (ca. 100 AD), Introductio Arithmetica P Let s(n) = djn;d<n d be the sum of the proper divisors of n. Abundant: s(n) > n. Deficient: s(n) < n. Perfect: s(n) = n. Ex: 5 is deficient (s(5) = 1), 12 is abundant (s(12) = 16), and 6 is perfect (s(6) = 6). 2 of 23 . In the case of those that are found between the too much and the too little, that is in equality, is produced virtue, just measure, propriety, beauty and things of that sort | of which the most exemplary form is that type of number which is called perfect.
    [Show full text]
  • ON CERTAIN P-ADIC BANACH LIMITS of P-ADIC TRIANGULAR MATRIX ALGEBRAS
    International Journal of Pure and Applied Mathematics Volume 85 No. 3 2013, 435-455 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: http://dx.doi.org/10.12732/ijpam.v85i3.1 ijpam.eu ON CERTAIN p-ADIC BANACH LIMITS OF p-ADIC TRIANGULAR MATRIX ALGEBRAS R.L. Baker University of Iowa Iowa City, Iowa 52242, USA Abstract: In this paper we investigate the class of p-adic triangular UHF (TUHF) Banach algebras.A p-adic TUHF Banach algebra is any unital p-adic Banach algebra of the form = n, where ( n) is an increasing sequence T T T T of p-adic Banach subalgebras of such that each n contains the identity of T S T T and is isomorphic as an Ωp-algebra to Tpn (Ωp) for some pn, where Tpn (Ωp) is the algebra of upper triangular pn pn matrices over the p-adic field Ωp. The main × result is that the supernatural number associated to a p-adic TUHF Banach algebra is an invariant of the algebra, provided that the algebra satisfies certain local dimensionality conditions. AMS Subject Classification: 12J25, 12J99, 46L99, 46S10 Key Words: p-adic Banach algebras, p-adic Banach limits, p-adic UHF algebras, p-adic Triangular UHF algebras 1. Introduction In the article, Triangular UHF algebras [2], the author of the present paper introduced a very concrete class of non-selfadjoint operator algebras which are currently called standard triangular UHF operator algebras. Briefly: a standard triangular UHF (TUHF) operator algebra is any unital Banach algebra that is isometrically isomorphic to a Banach algebra inductive limit of the form = T lim(Tp ; σp p ).
    [Show full text]
  • MATHEMATICS Algebra, Geometry, Combinatorics
    MATHEMATICS Algebra, geometry, combinatorics Dr Mark V Lawson October 24, 2014 ii Contents 1 The nature of mathematics 1 1.1 What are algebra, geometry and combinatorics? . .1 1.1.1 Algebra . .1 1.1.2 Geometry . .5 1.1.3 Combinatorics . .7 1.2 The scope of mathematics . .8 1.3 Pure versus applied mathematics . .9 1.4 The antiquity of mathematics . 11 1.5 The modernity of mathematics . 12 1.6 The legacy of the Greeks . 14 1.7 The legacy of the Romans . 15 1.8 What they didn't tell you in school . 15 1.9 Further reading and links . 16 2 Proofs 19 2.1 How do we know what we think is true is true? . 20 2.2 Three fundamental assumptions of logic . 22 2.3 Examples of proofs . 23 2.3.1 Proof 1 . 23 2.3.2 Proof 2 . 26 2.3.3 Proof 3 . 28 2.3.4 Proof 4 . 29 2.3.5 Proof 5 . 31 2.4 Axioms . 37 2.5 Mathematics and the real world . 41 2.6 Proving something false . 41 2.7 Key points . 42 2.8 Mathematical creativity . 43 i ii CONTENTS 2.9 Set theory: the language of mathematics . 43 2.10 Proof by induction . 52 3 High-school algebra revisited 57 3.1 The rules of the game . 57 3.1.1 The axioms . 57 3.1.2 Indices . 63 3.1.3 Sigma notation . 66 3.1.4 Infinite sums . 68 3.2 Solving quadratic equations . 70 3.3 Order . 76 3.4 The real numbers .
    [Show full text]
  • Relative Abelian Kernels of Some Classes of Transformation Monoids
    BULL. AUSTRAL. MATH. SOC. 20M05, 20M20, 20K45, 20MO7 VOL. 73 (2006) [375-404] RELATIVE ABELIAN KERNELS OF SOME CLASSES OF TRANSFORMATION MONOIDS E. CORDEIRO, M. DELGADO AND V.H. FERNANDES We consider the symmetric inverse monoid Zn of an n-element chain and its in- verse submonoids VOIn, VOVIn, VOVln and VO1Hn of all order-preserving, order-preserving or order-reversing, orientation-preserving and orientation-preserving or orientation-reversing transformations, respectively, and give descriptions of their Abelian kernels relative to decidable pseudovarieties of Abelian groups. INTRODUCTION The computability of kernels of finite monoids became a popular problem in Finite Semigroup Theory after a paper of Rhodes and Tilson [23]. Independent solutions were given by Ash [3] and Ribes and Zalesskii [24]. Both solutions contains deep results which led to the development of the theory in various directions. See for instance [19, 2]. Com- puting kernels relative to other pseudovarieties had then interested various researchers; for examples, see Ribes and Zalesskii [25], the second author [5] or Steinberg [26]. As computing kernels are directly connected with the Mal'cev product of pseudovarieties of monoids where the second factor is a pseudovariety of groups (see [18]), decidability results can be obtained in this way. The interest of the second author in using in practice the algorithm to compute the Abelian kernel of a finite monoid obtained in [5] led him to work towards a concrete implementation. Having this idea in mind, by detailing some aspects and improving the efficiency of that algorithm (see [6]) a concrete implementation in GAP [27] was then produced.
    [Show full text]