DOCSLIB.ORG

Algebra and Number Theory Lecture Notes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Algebra and Number Theory Lecture Notes ∗ Algebra and Number Theory Lecture Notes Szymon Brzostowski December 5, 2013 This text is intended to be a more detailed, or rather – a more intuitive, version of the lecture itself. In order to get the most of it, I recommend reading it with your class notes at hand. Hopefully, this should allow for full comprehension of the material given. Notation. The symbol „ ” will denote an assignment, for example a 2 means that the name a carries the value 2. More generally, „ ” can be treated as a defi- nition of a new concept (on its left side) using already known concepts (appearing on its right side). We will also abbreviate the phrase „if and only if” as „iff”. Table of contents 1 Numbers and operations ................................. 3 2 Which numbers are „best”? ............................... 5 3 The first abstraction – the concept of a group .................. 7 4 ...Divisibility in Z ...................................... 9 5 ...Second abstraction – the concept of a ring .................. 11 6 The concept of an isomorphism ........................... 12 7 The direct sum ....................................... 15 8 The third abstraction – the concept of a field ................. 17 9 Congruences ......................................... 18 10 GCD ............................................. 20 ∗ . This document has been written using the GNU TEXMAC S text editor (see www.texmacs.org). 2 Szymon Brzostowski 11 Solving a x b and a x = b ............................ 23 · ≡ n · n 12 The group G n and the ϕ function ......................... 26 13 Fermat’s little theorem and its generalizations ................ 29 14 Chinese remainder theorem ............................. 34 15 The prime numbers ................................... 37 16 Systems of linear equations ............................. 41 16.1 Fromonetoseveralequations . ..... 41 16.2 Triangular systems and back substitution . ........ 44 16.3 Matrixnotationforalinearsystem . ...... 45 16.4 Rowechelonformofamatrix ........................ 49 Further reading ......................................... 49 1. Numbers and operations 3 1 Numbers and operations We begin by setting the terminology for the basic number sets: the naturals (=positive integers): N 1 , 2, 3, , → { } the integers : Z 0, 1 , 2, 3, , → { ± ± ± } p Q Z N the rationals : q : p , q , → n ∈ ∈ o √ the reals : R; it is a set larger than Q; it contains e.g. the radicals ( √2 , 3 , 10√10 → 3 √5 4 +√ 7 , ) and also other numbers, like p, e, p + 1 , . p Note that by convention 0 N. The symbol N will denote the set N 0 . 0 ∪ { } There is one more set of numbers, easily constructed from R: the complex numbers : C a + b i: a, b R . Here two complex numbers are → { ∈ } added and multiplied similarly as polynomials, but further simplification of the result is possible because the new number „i” satisfies i2 =ii= 1 . − Clearly, we have the following chain of inclusions: N $ Z $ Q $ R $ C. Example. 1. Here’s how you add two complex numbers: (2+i)+(1 i) = 2+i+1 i = 3+(1 1) i=3+0 i = 3. − skip the brackets − collect similar terms − · · 2. And here is multiplication: (2+i) (1 i) = 2 (1 i)+i (1 i) = · − use the distributive law · − · − = 2 2 i + 1 i i i = distributive law again − · · − · = 2 + ( 2+1) i ( 1) = 3 i. use the relation for i2 − · − − collect similar terms − Analyzing the first example above, it is not hard to produce a general rule for the addition of complex numbers: Addition Rule in C. ( a + b i) + ( c + d i) ( a + c) + ( b + d) i. a,b,c,d R ∀ ∈ (The universal quantifier is read „for all”.) ∀ A similar formula for the product of two complex numbers holds ( try to find it yourself! ): Multiplication Rule in C. ( a + b i) ( c + d i) ( a c b d) + ( a d + b c) i. a,b,c,d R · · − · · · · ∀ ∈ 4 Szymon Brzostowski Observe that thanks to using symbols instead of specific numbers, above there are formulas that work for every choice of (real) values of a,b,c,d. So although it is enough How for you to know how to calculate with complex numbers to arrive at the correct result, Algebra these formulas give you the result almost automatically once you remember them. In works algebra you generally work with symbols and you calculate using a certain set of rules for manipulating the symbols. Some of these rules you should already be familiar with. The basic rules for computations with numbers are 2+1=1+2, 1 + 1 = 1 + 1 , and – in symbols – → 2 3 3 2 a + b = b + a ( commutativity ) (This law allows you to change the order of the summands.) ( 1+2)+ 3 = 1 + 3 = 5 = 1 + 7 = 1 + 2 + 3 , and – in symbols – → − 2 2 2 − 2 − 2 a + ( b + c) = ( a + b) + c ( associativity ) (This law allows you to skip all brackets when adding – the result is always the same regardless of which numbers you add first.) (2 + 1) 1 = 3 1 = 1 = 2 + 1 = 2 1 + 1 1 , and – in symbols – → · 3 · 3 3 3 · 3 · 3 ( a + b) c = a c + b c ( distributivity ) · · · (This law allows you to distribute multiplication over addition.) Note that the first two rules above are also true when multiplying numbers (i.e. with + replaced by ). This suggests that perhaps such rules could be interpreted more · abstractly. Namely, the symbol + (or ) might denote an unspecified binary operation. · Of course, in such generality it might be appropriate to use some new symbols (like , ⊕ , , ⊞ or ) for the binary operations and sometimes it is done so, especially when ⊙ ◦ there is a danger of some confusion. Observe however, that in the case of complex numbers that we have considered, we used the ordinary symbols + and , rather than · some new ones, and we called them „addition” and „multiplication”, respectively. This is natural, because those operations are in fact extensions of the usual operations of addition and multiplication from the real numbers to a wider domain (this means that the „new” binary operations work in the „old” way for those complex numbers which are real, e.g. (2 +0i)+(3 +0i)=5 + 0 i and similarly (2 + 0 i) (3 + 0 i) = 6 + 0 i). Also in · computer science, a common practice is to implement some new binary operations and still denote them by + and although in fact they overload or even totally redefine the · standard operations of addition and multiplication. Example. One can define a truncated subtraction in N0 by the formula 0, if a < b a + b . a b, if a > b − There is everything in order with such definition, but you should feel that the usage of + is a little cumbersome here (after all, we are rather subtracting than adding). In elementary arithmetic this truncated subtraction is usually denoted by „ . ”. − 2. Which numbers are „best”? 5 2 Which numbers are „best”? T here is no unique answer to such question, because it depends on the properties you are interested in. For example, testing associativity and commutativity gives N Z Q R C Ass. YES YES YES YES YES + Com. YES YES YES YES YES , Ass. YES YES YES YES YES · Com. YES YES YES YES YES so in these respects there is no apparent difference between N, Z, Q, R and C. Actually, in principle , in order to verify that the entries in the above table are correct, it would be enough to check these properties for C because then they are automatically valid for the smaller sets N, Z, Q and R. In practice , this is done the other way around because of the order in which the number sets are actually constructed ( N Z Q R C). → → → → As an example of such verification, try: Exercise 1. Check that + and are associative and commutative in C (assuming you know this · for R!) using the formal definitions of + and given on page 3. · Commentary. Proving the usual empirical properties of the operations + and is not · totally trivial, even for N. First, one needs to properly define the set N (this can be done using Set Theory) and the operations + and . Only then is it possible to actually prove · their properties (this is done in a branch of mathematics called Theoretical Arithmetic). Fortunately, you have so much empirical evidence for their validity from everyday life, you should not feel uncomfortably using those properties. G oing back to our question again: are the various sets of numbers essentially different? Yes, they are... But only when you start thinking about solving equations in them. Suppose we want to solve the equation 2 + x = 1 in the naturals. Clearly, this is impossible because we have x N 1 2 3 ∈ a result always greater than 1 ! 2 + x 3 4 5 But the same equation has a (unique!) solution x = 1 in Z. A difference! − The following table shows which equations can be solved in the various number sets: Equation to solve N Z Q R C a + x = b NO YES YES YES YES a x = b, a 0 NO NO YES YES YES . (1) · x2 = 2 NO NO NO YES YES x2 = 1 NO NO NO NO YES − As we can see, Q is different than Z if considered with multiplication (e.g. 2 x = 1 · Z 2 3 has no solutions in ; on the other hand the equation 3 x= 5 does have the solution 9 · − x= in Q). Also, C is „better” than the other sets with respect to . But what is − 10 · the difference between Q and R? Part of the answer is shown in the above table. The full answer is beyond the scope of Algebra: the real difference is that in R there exist „limits” for all the sequences that „should have” a limit. The same property is true for C. Hence, R and C have different analytic features than Q. 6 Szymon Brzostowski 1 1 1 1 1 1 , , , Q Example.
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]