DOCSLIB.ORG

RATIONAL and INTEGRAL POINTS on CURVES Andrew Granville

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
RATIONAL and INTEGRAL POINTS on CURVES Andrew Granville RATIONAL AND INTEGRAL POINTS ON CURVES Andrew Granville Abstract. These are notes from a course given in Autumn of 2011 at the Universit´ede Montr´eal. I have written things fairly completely up to about chapter 15. There is a lot more to be added on elliptic curves, and then on higher degree curves. Also stuff on integral points via Diophantine approximation. Please email me with any corrections or remarks ([email protected]). Table of Contents Preface. Diophantine equations and curves Chapter 1. Linear equations 1.1. Linear equations in two unknowns 1.2. The Euclidean algorithm 1.3. Several simultaneous linear equations in several unknowns 1.4. Rational solutions and lattices Chapter 2. Some useful mathematics 2.1. Rational approximationsp to real numbers 2.2. The Irrationality of d. 2.3. Continued fractions for real numbers 2.4. Solving the cubic. Pre-cursors to Galois theory. 2.5. Constructibility and Galois theory 2.6. Resultants and Discriminants 2.7. M¨obiustransformations: Lines and circles go to lines and circles 2.8. Groups 2.9. Ideals Chapter 3. Quadratic equations 3.1. The Pythagorean equation 3.2. Fermat's “infinite descent" 3.3. Squares mod m 3.4. The law of quadratic reciprocity 3.5. Sums of two squares 3.6. The values of x2 + dy2 Typeset by AMS-TEX 2 3.7. Solutions to quadratic equations 3.8. Pell's equation 3.9. Descent on solutions of x2 − dy2 = n; d > 0 Chapter 4. Binary quadratic forms 4.1. Representation of integers by binary quadratic forms 4.2. Equivalence classes of binary quadratic forms 4.3. Class number one 4.4. Ideals in quadratic fields 4.5. Composition laws 4.6. Bhargava composition 4.7. The idoneal numbers 4.8. SL(2; Z)-transformations. Forms-Ideals-Transformations 4.9. The ring of integers of a quadratic field, revisited Chapter 5. Real quadratic fields 5.1. Quadratic irrationals and periodic continued fractions 5.2. Pell's equation 5.3. The size of solutions to Pell's equation 5.4. More examples of Pell's equation 5.5. Binary quadratic forms with positive discriminant, and continued fractions Chapter 6. L-functions and class numbers 6.1. Counting solutions to Pell's equation (a heuristic) 6.2. Dirichlet's class number formula 6.3. The class number one problem in real quadratic fields 6.4. Dirichlet L-functions Chapter 7. More about quadratic forms and lattices 7.1. Minkowski and lattices 7.2. The number of representations as the sum of two squares 7.3. The number of representations by arbitrary binary quadratic forms 7.4. The number of representations as the sum of three squares 7.5. The number of representations as the sum of four squares 7.6. Universality of quadratic forms 7.7. Representation by positive definite quadratic forms 7.8. Descent and the quadratics. Chapter 8. More useful mathematics 8.1. Finite fields 8.2. Affine vs. Projective 8.3. Lifting solutions Chapter 9. Counting points mod p 3 9.1. Linear and quadratic equations mod p 9.2. The diagonal cubic equation mod p 9.3. The equation y2 = x3 + ax 9.4. The equation y2 = x3 + b 9.5. Counting solutions mod p 9.6. Degenerate elliptic curves mod p 9.7. Lifting solutions 9.8. Simultaneous Pell equations Chapter 10. Power series, partitions, and magical modularity 10.1. Partitions 10.2. The numerology of partitions 10.3. Generating functions for binary quadratic forms, and L-functions 10.4. Poisson's summation formula and the theta function 10.5. Jacobi's powerful triple product identity 10.6. An example of modularity: y2 = x3 − x Chapter 11. Degree three curves | why the curve y2 = x3 + ax + b? 11.1. Parametric families of rational points on curves 11.2. Constructing new rational points on cubic curves from old ones 11.3. Cubic curves into Weierstrass form 11.4. Diagonal cubic curves 11.5. y2 equals a quartic with a rational point 11.6. The intersection of two quadratic polynomials in three variables (that is, the intersection of two quadratic surfaces) 11.7. Doubling a point on a diagonal cubic Chapter 12. The group of rational points on an elliptic curve 12.1. The group of rational points on an elliptic curve 12.2. The group law on the circle, as an elliptic curve 12.3. No non-trivial rational points by descent 12.4. The group of rational points of y2 = x3 − x 12.5. The arithmetic of a torsion point 12.6. Embedding torsion in Fp 12.7. The size of rational points Chapter 13. Mordell's theorem { E(Q) is finitely generated 13.1. The proof of Mordell's Theorem over the rationals 13.2. Another example: Four squares in an arithmetic progression 13.3. Mordell's Theorem in number fields 13.4. More precise bounds on naive height 13.5. N´eron-Tate height 13.6. An inner product 13.7. The number of points in the Mordell-Weil lattice up to height x 4 13.8. Magic squares and elliptic curves Chapter 14. The Weierstrass }-function, and endomorphisms of elliptic curves 14.1. Double periodicity and the Weierstrass }-function. 14.2. Parametrizing elliptic curves 14.3. Torsion points in C 14.4. The converse theorem 14.5. Maps between elliptic curves over C. Complex multiplication. 14.6. Homomorphisms and endomorphisms for elliptic curves 14.7. The Frobenius map and the number of points on E(Fpk ) 14.8. Classifying endomorphisms when t =6 0 14.9. Torsion points in Fp Chapter 15. Modular Forms 15.1. The magic of Eisenstein series 15.2. The Fourier expansion of an Eisenstein series 15.3. Modular forms 15.4. Determining spaces of modular forms 15.5. The j-function 15.6. The j-invariant and complex multiplication 15.7. Almost an Eisenstein series 15.8. Sublattice and subgroups 15.9. Hecke operators 15.10. The Mellin transform and the construction of L-functions 15.11. Congruence subgroups Chapters 16, 17, 18. More stuff on elliptic curves Chapter 19. Integral points on elliptic curves 19.1. Sums of two cubes 19.2. Taxicab numbers and other diagonal surfaces Chapter 20. Integral points on higher genus curves 20.1. Thue's Theorem Chapter 21. Diophantine equations in polynomials Chapter 22. Fermat's Last Theorem 21.1. FLT and Sophie Germain 21.2. The abc-conjecture 21.3. Faltings' Theorem n´eeMordell's conjecture 5 Preface It is natural for a mathematician to ask for the solutions to a given equation. To un- derstand the solutions in complex numbers may require ideas from analysis; to understand the solutions in integers requires ideas from number theory. The main difference is that, in the complex numbers, one can solve equations by using limits. For example to find a complex number x for which x2 = 2p we need to find increasingly good approximations to a solution, in order to prove that 2, the limit of the approximations, exists. However to prove that there is no rational number p=q such that (p=q)2 = 2 takes rather different techniques, and these are what we focus on in these notes. Finding integral or rational solutions to a polynomial equation (or finite set of poly- nomial equations) is known as Diophantine arithmetic in honour of Diophantus,1 a Greek mathematician who lived in Alexandria in the third century A.D.2 His thirteen volume Arithmetica dealt with solutions to equations in integers and rationals,3 which are now known as Diophantine equations. Typically the subject of Diophantine equations is motivated by the quest to prove that there are no integer solutions to certain Diophantine equations. For example, Catalan's 1844 conjecture that the only integer solution to xp − yq = 1 with x; y ≥ 1 and p; q ≥ 1 is 32 − 23 = 1; which was proved by Preda Mih˘ailescuin 2002. Also Fermat's assertion from around 1637, known as \Fermat's Last Theorem",4 that there are no integers x; y; z ≥ 1 and n ≥ 3 for which xp + yp = zp; which was finally proved in 1995 by Sir Andrew Wiles. Our goals will not be so elevated. We will mostly focus on the beautiful theory for trying to find rational solutions to Diophantine equations in just two variables, of degrees one, two and three. Degree one equations are more-or-less completely understood, but there are fundamental questions for degrees two and three that remain open, and indeed understanding degree three equations is perhaps the most researched problem in modern number theory. We shall assume that the reader is familiar with a first course in number theory. When we require more advanced concepts we shall explain them. 1Bachet's 1621 Latin translation of Diophantus's Arithmetica helped spark the mathematical Renaissance. 2Many of the most famous ancient \Greek mathematicians", such as Diophantus, Eratosthenes, Euclid, Hero, Pappus, and Ptolemy, actually worked in Alexandria, Egypt. 3Though only parts of six of the volumes have survived. 4Indeed he wrote \it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain" in his copy of Arithmetica.A copy of Arithmetica with Fermat's notes was published after his death, by his son in 1670. 1 1. Linear equations 1.1. Linear equations in two unknowns. The simplest Diophantine problem one encounters is to ask whether, given non-zero integers a; b and integer c, there are solutions in integers m; n to (1.1) am + bn = c : One encounters this problem in a first course in number theory, and proves that there are such solutions if and only if gcd(a; b) divides c.
Load more

Recommended publications
  • BLECHER • All Numbers Below Are Integers, Usually Positive. • Read Up
    MATH 4389{ALGEBRA `CHEATSHEET'{BLECHER • All numbers below are integers, usually positive. • Read up on e.g. wikipedia on the Euclidean algorithm. • Diophantine equation ax + by = c has solutions iff gcd(a; b) doesn't divide c. One may find the solutions using the Euclidean algorithm (google this). • Euclid's lemma: prime pjab implies pja or pjb. • gcd(m; n) · lcm(m; n) = mn. • a ≡ b (mod m) means mj(a − b). An equivalence relation (reflexive, symmetric, transitive) on the integers. • ax ≡ b (mod m) has a solution iff gcd(m; a)jb. • Division Algorithm: there exist unique integers q and r with a = bq + r and 0 ≤ r < b; so a ≡ r (mod b). • Fermat's theorem: If p is prime then ap ≡ a (mod p) Also, ap−1 ≡ 1 (mod p) if p does not divide a. • For definitions for groups, rings, fields, subgroups, order of element, etc see Algebra fact sheet" below. • Zm = f0; 1; ··· ; m−1g with addition and multiplication mod m, is a group and commutative ring (= Z =m Z) • Order of n in Zm (i.e. smallest k s.t. kn ≡ 0 (mod m)) is m=gcd(m; n). • hai denotes the subgroup generated by a (the smallest subgroup containing a). For example h2i = f0; 2; 2+2 = 4; 2 + 2 + 2 = 6g in Z8. (In ring theory it may be the subring generated by a). • A cyclic group is a group that can be generated by a single element (the group generator). They are abelian. • A finite group is cyclic iff G = fg; g + g; ··· ; g + g + ··· + g = 0g; so G = hgi.
    [Show full text]
  • Countable Torsion-Free Abelian Groups
    COUNTABLE TORSION-FREE ABELIAN GROUPS By M. O'N. CAMPBELL [Received 27 January 1959.—Read 19 February 1959] Introduction IN this paper we present a new classification of the countable torsion-free abelian groups. Since any abelian group 0 may be regarded as an extension of a periodic group (namely the torsion part of G) by a torsion-free group, and since there exists the well-known complete classification of the count- able periodic abelian groups due to Ulm, it is clear that the classification problem studied here is of great interest. By a group we shall always mean an abelian group, and the additive notation will be employed throughout. It is well known that all the maximal linearly independent sets of elements of a given group are car- dinally equivalent; the corresponding cardinal number is the rank of the group. Derry (2) and Mal'cev (4) have studied the torsion-free groups of finite rank only, and have given classifications of these groups in terms of certain equivalence classes of systems of matrices with £>-adic elements. Szekeres (5) attempted to abolish the finiteness restriction on rank; he extracted certain invariants, but did not derive a complete classification. By a basis of a torsion-free group 0 we mean any maximal linearly independent subset of G. A subgroup generated by a basis of G will be called a basal subgroup or described as basal in G. Thus U is a basal sub- group of G if and only if (i) U is free abelian, and (ii) the factor group G/U is periodic.
    [Show full text]
  • Arxiv:1909.05807V1 [Math.RA] 12 Sep 2019 Cs T
    MODULES OVER TRUSSES VS MODULES OVER RINGS: DIRECT SUMS AND FREE MODULES TOMASZ BRZEZINSKI´ AND BERNARD RYBOLOWICZ Abstract. Categorical constructions on heaps and modules over trusses are con- sidered and contrasted with the corresponding constructions on groups and rings. These include explicit description of free heaps and free Abelian heaps, coprod- ucts or direct sums of Abelian heaps and modules over trusses, and description and analysis of free modules over trusses. It is shown that the direct sum of two non-empty Abelian heaps is always infinite and isomorphic to the heap associated to the direct sums of the group retracts of both heaps and Z. Direct sum is used to extend a given truss to a ring-type truss or a unital truss (or both). Free mod- ules are constructed as direct sums of a truss. It is shown that only free rank-one modules are free as modules over the associated truss. On the other hand, if a (finitely generated) module over a truss associated to a ring is free, then so is the corresponding quotient-by-absorbers module over this ring. 1. Introduction Trusses and skew trusses were defined in [3] in order to capture the nature of the distinctive distributive law that characterises braces and skew braces [12], [6], [9]. A (one-sided) truss is a set with a ternary operation which makes it into a heap or herd (see [10], [11], [1] or [13]) together with an associative binary operation that distributes (on one side or both) over the heap ternary operation. If the specific bi- nary operation admits it, a choice of a particular element could fix a group structure on a heap in a way that turns the truss into a ring or a brace-like system (which becomes a brace provided the binary operation admits inverses).
    [Show full text]
  • Groups with Preassigned Central and Central Quotient Group*
    GROUPS WITH PREASSIGNED CENTRAL AND CENTRAL QUOTIENT GROUP* BY REINHOLD BAER Every group G determines two important structural invariants, namely its central C(G) and its central quotient group Q(G) =G/C(G). Concerning these two invariants the following two problems seem to be most elementary. If A and P are two groups, to find necessary and sufficient conditions for the existence of a group G such that A and C(G), B and Q(G) are isomorphic (existence problem) and to find necessary and sufficient conditions for the existence of an isomorphism between any two groups G' and G" such that the groups A, C(G') and C(G") as well as the groups P, Q(G') and Q(G") are isomorphic (uniqueness problem). This paper presents a solution of the exist- ence problem under the hypothesis that P (the presumptive central quotient group) is a direct product of (a finite or infinite number of finite or infinite) cyclic groups whereas a solution of the uniqueness problem is given only un- der the hypothesis that P is an abelian group with a finite number of genera- tors. There is hardly any previous work concerning these problems. Only those abelian groups with a finite number of generators which are central quotient groups of suitable groups have been characterized before, t 1. Before enunciating our principal results (in §2) some notation and con- cepts concerning abelian groups will have to be recorded for future reference. If G is any abelian group, then the composition of the elements in G is denoted as addition: x+y.
    [Show full text]
  • Group Theory
    Algebra Math Notes • Study Guide Group Theory Table of Contents Groups..................................................................................................................................................................... 3 Binary Operations ............................................................................................................................................................. 3 Groups .............................................................................................................................................................................. 3 Examples of Groups ......................................................................................................................................................... 4 Cyclic Groups ................................................................................................................................................................... 5 Homomorphisms and Normal Subgroups ......................................................................................................................... 5 Cosets and Quotient Groups ............................................................................................................................................ 6 Isomorphism Theorems .................................................................................................................................................... 7 Product Groups ...............................................................................................................................................................
    [Show full text]
  • Countable Torsion Products of Abelian /^-Groups E
    proceedings of the american mathematical society Volume 37, Number 1, January 1973 COUNTABLE TORSION PRODUCTS OF ABELIAN /^-GROUPS E. L. LADY Abstract. Let Au A2, • ■■ be direct sums of cyclic/j-groups, and G=r]^7" An be the torsion subgroup of the product. The product decomposition is used to define a topology on G in which each G [pT]is complete and the restriction to G[pr] of any endomorphism is continuous. Theorems are then derived dual to those proved by Irwin, Richman and Walker for countable direct sums of torsion complete groups. It is shown that every subgroup of G[p] supports a pure subgroup of G, and that if G=H®K, then H^tTTy Gn, where the G„ are direct sums of cyclics. A weak isomorphic refine- ment theorem is proved for decompositions of G as a torsion product. Finally, in answer to a question of Irwin and O'Neill, an example is given of a direct decomposition of G that is not induced by a decomposition of Y~[y A„. 1. The product topology. Let G=tT~[x An be the torsion subgroup of \~[y An, where the An are abelian /»-groups. For each nonnegative integer n, let Gn=t \~jr> n A{, considered as a subgroup of G in the natural way. The product topology on G is defined by taking the Gn to be a neigh- borhood basis at 0. Thus the product topology is the topology that t \~\x An inherits as a subspace of the cartesian product of the An, where these are considered discrete.
    [Show full text]
  • Admissible Direct Decompositions of Direct Sums of Abelian Groups of Rank One
    Separatunt PUBLICATION ES MATHEHATICAE TOMUS 10. FASC. DEBRECEN 1963 FUNDAVERUNT: A. O M T. SZELE ET O. VARGA COMMI SSI O REDACTORUM: J. ACZÉL, B. GY1RES, A. KERTÉSZ, A. RAPCSAK, A. RÉNYI, ET O. VARGA SECRETARIUS COMMI SSI ON'S: B. BARNA L. G. Kovics Admissible direct decompositions of direct sums o f abelian groups o f rank one INSTITUTUM M AT H EM AT IC U M U N IVER SITATIS DEBRECENIENSIS HUNGARIA 254 Admissible direct decompositions of direct sums of abelian groups of rank one By L. G. KOVAC S (Manchester) The starting point o f the theory o f ordinary representations is MASCHKE'S Theorem, which states that every representation of a finite group over a field whose characteristic does not divide the order o f the group is completely reducible (see e. g. VAN DER WAERDEN [6], p. 182). A partial generalization of this theorem has recently been given by O. GRÜN in [2], and the main step of the classical proof of the theorem has been generalized by M. F. NEWMAN and the author in [4]. Both of these results arose out of a shift in the point of view: they do not refer to repre- sentations, but to direct decompositions of abelian groups, admissible with respect to a finite group of operators (in the sense of KUROSH [5], § 15). The aim o f this paper is to present an extension of GRON's result, exploiting the start made in [4]. The terminology follows, apart from minor deviations, that o f FUCHS'S book [1].
    [Show full text]
  • Have Talked About Since the First Exam Normal Subgroups. the Integers
    Math 417 Group Theory Second Exam Things we (will) have talked about since the first exam Normal subgroups. The integers mod n can be viewed either as having elements {0, 1,...,n− 1}, or the elements are equivalence classes of integers, where a ∼ b if n|b − a. In the latter case, we add elements by adding representatives of equivalence classes (and then need to show that the resulting ‘element’ is independent of the choices made). This notion can be generalized: For G a group and H ≤ G a subgroup, defining a multiplication on (left) cosets by (aH)(bH)= (ab)H requires, to be well-defined, that a1H = a2H and b1H = b2H implies (a1b1)H = −1 −1 −1 −1 −1 2 −1 (a2b2)H. That is, we need a1 a2, b1 b2 ∈ H implies (a1b1) (a2b2) = b1 (a1 a )b1(b1 b2) is −1 −1 2 −1 in H. This then requires b1 (a1 a )b1 = b1 hb1 ∈ H for every b1 ∈ G and h ∈ H. So we define: H ≤ G is a normal subgroup of G if g−1Hg = {g−1hg : h ∈ H} = H for every g ∈ G. Then the multiplication above makes the set of (left) cosets G/H = {gH : g ∈ G} a group. −1 −1 H = eGH is the identity element, and (gH) = g H. The order of G/H is the number of cosets of H in G, i.e., the index [G : H] of H in G. We use H⊳G to denote “H is normal in G”. We call G/H the quotient of G by H.
    [Show full text]
  • Primary Decomposition Theorem for Finite Abelian Groups
    Subject : MATHEMATICS Paper 1 : ABSTRACT ALGEBRA Chapter 1 : Direct Product of Groups Module 3 : Primary decomposition theorem for finite abelian groups Anjan Kumar Bhuniya Department of Mathematics Visva-Bharati; Santiniketan West Bengal 1 Primary decomposition theorem for finite abelian groups Learning outcomes: 1. p-primary abelian groups. 2. Primary Decomposition Theorem. 3. p-primary components. In this module we prove that every finite abelian group can be expressed as an internal direct product of a family of p-primary subgroups. In most of the results on the structure of finite abelian groups the feature that plays the most powerful role is that every finite abelian group is a Z-module. So, it is convenient to consider addition as the binary operation in the study of finite abelian groups. Internal direct product of a family of subgroups G1;G2; ··· ;Gk of a group G, if it is abelian, is called direct sum of G1;G2; ··· ;Gk and is denoted by G1 ⊕ G2 ⊕ · · · ⊕ Gk. Thus a finite abelian group G is a direct sum of subgroups G1;G2; ··· ;Gk if and only if G = G1 + G2 + ··· + Gk and Gi \(G1 +···+Gi−1 +Gi+1 +···+Gk) = f0g for every i = 1; 2; ··· ; k. In view of the fact that every internal direct sum of the subgroups G1;G2; ··· ;Gk is isomorphic to the external direct product of G1;G2; ··· ;Gk, we call both the internal direct sum and the external direct product simply as direct sum and use the notation G1 ⊕G2 ⊕· · ·⊕Gk to represent direct sum of groups G1;G2; ··· ;Gk. Definition 0.1.
    [Show full text]
  • The Characterization of a Line Arrangement Whose Fundamental Group of the Complement Is a Direct Sum of Free Groups
    Algebraic & Geometric Topology 10 (2010) 1285–1304 1285 The characterization of a line arrangement whose fundamental group of the complement is a direct sum of free groups MEITAL ELIYAHU ERAN LIBERMAN MALKA SCHAPS MINA TEICHER Kwai Man Fan proved that if the intersection lattice of a line arrangement does not contain a cycle, then the fundamental group of its complement is a direct sum of infinite and cyclic free groups. He also conjectured that the converse is true as well. The main purpose of this paper is to prove this conjecture. 14F35, 32S22; 14N30, 52C30 1 Introduction An arrangement of lines is a finite collection of C–affine subspaces of dimension 1. For such an arrangement † C2 , there is a natural projective arrangement † of  lines in CP 2 associated to it, obtained by adding to each line its corresponding point at infinity. The problem of finding connections between the topology of C2 † and the combinatorial theory of † is one of the main problems in the theory of line arrangements (see for example Cohen and Suciu [3]). The main motivations for studying the topology of C2 † are derived from the areas of hypergeometric functions, singularity theory and algebraic geometry. Given an arrangement †, we define the graph G.†/ in the way it is defined by Fan [7]. We first connect all higher multiple points of † which lie on a line Li by an arbitrary simple arc ˛i Li which does not go through any double point of †. Taking the union of the set of all higher multiple points and all of this simple arcs, we obtain G.†/.
    [Show full text]
  • Self-Small Abelian Groups
    Bull. Aust. Math. Soc. 80 (2009), 205–216 doi:10.1017/S0004972709000185 SELF-SMALL ABELIAN GROUPS ULRICH ALBRECHT ˛, SIMION BREAZ and WILLIAM WICKLESS (Received 23 October 2008) Abstract This paper investigates self-small abelian groups of finite torsion-free rank. We obtain a new characterization of infinite self-small groups. In addition, self-small groups of torsion-free rank 1 and their finite direct sums are discussed. 2000 Mathematics subject classification: primary 20K21. Keywords and phrases: self-small group, projection condition, essentially W-reduced group. 1. Introduction It is the goal of this paper to investigate self-small abelian groups. Introduced by Arnold and Murley [6], these are the abelian groups G such that, for every index .I / L set I and every α 2 Hom.G; G /, there is a finite subset J ⊆ I with α.G/ ⊆ J G. Since every torsion-free abelian group of finite rank is self-small, and all self-small torsion groups are finite by [6, Proposition 3.1], our investigation focuses on mixed groups. Section 2 introduces several important classes of self-small mixed groups, and discusses their basic properties. In Section 3 we give a new characterization of infinite self-small groups of finite torsion-free rank (Theorem 3.1), while Section 4 extends the notion of completely decomposable groups from the torsion-free case to the case of mixed groups. Although we use the standard notation from [12], some terminology shall be mentioned for the benefit of the reader. A mixed abelian group G is honest if tG L is not a direct summand of G, where tG D p G p denotes the torsion subgroup of G, and G p is its p-torsion subgroup.
    [Show full text]
  • Abstract Algebra
    Abstract Algebra Groups, Rings and Fields, Advanced Group Theory, Modules and Noetherian Rings, Field Theory YOTSANAN MEEMARK Semi-formal based on the graduate courses 2301613–4 Abstract Algebra I & II, offered at Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University Published by Yotsanan Meemark Department of Mathematics and Computer Science Faculty of Science, Chulalongkorn University, Bangkok, 10330 Thailand First digital edition October 2013 First bound edition May 2014 Second bound edition August 2015 Available for free download at http://pioneer.netserv.chula.ac.th/~myotsana/ Please cite this book as: Y. Meemark, Abstract Algebra, 2015, PDF available at http://pioneer.netserv.chula.ac.th/~myotsana/ Any comment or suggestion, please write to [email protected] c 2015 by Yotsanan Meemark. Meemark, Yotsanan Abstract Algebra / Yotsanan Meemark – 2nd ed. Bangkok: Danex Intercorporation Co., Ltd., 2015. 195pp. ISBN 978-616-361-389-9 Printed by Danex Intercorporation Co., Ltd., Bangkok, Thailand. www.protexts.com Foreword This book is written based on two graduate abstract algebra courses offered at Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University. It grows out of my lecture notes that I used while I was teaching those courses many times. My intention is to develop essential topics in algebra that can be used in research as illustrated some in the final chapter. Also, it can be served as a standard reference for preparing for a qualifying examination in Algebra. I have tried to make it self-contained as much as possible. However, it may not be suitable for reading it for the first course in abstract algebra.
    [Show full text]