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97358 TOPICS IN ALGEBRA I. N. HERSTEIN University of Chicago GINN AND COMPANY A Xerox Company WALTHAM, MASSACHUSETTS - TORONTO - LONDON To MARIANNE Copyright O) 1964, by Ginn and Company AU Rights Reserved LIBRABY OF CONGRESS CATALOG CARD NUMBER: 63-17982 vi PREFACE chapter on rings, the two-square theorem of Fermat is exhibited as a direct consequence of the theory developed for Euclidean rings. The subject matter chosen for discussion has been picked not only be- cause it has become standard to present it at this level or because it is important in the whole general development but also with an eye to this “conereteness.” For this reason I chose to omit the Jordan-Hólder theovema, which certainly could have easily been included in the results derived about groups. However, to appreciate this result for its own sake requires a great deal of hindsight and to see it used effectively would require too great a digression. True, one could develop the whole theory of dimension of a vector space as one of its corollaries, but, for the first time around, this seems like a much too faney and unnatural approach to something so basic and down-to-earth. Likewise, there is no mention of tensor prod- ucts or related constructions. There is so much time and opportunity to become abstract; shy rush it at the beginning? A word about the problems. There are a great number of them. Tt would be an extraordinary student indeed who could solve them all. Some are present merely to complete proofs in the text material, others to illustrate and to give practice in the results obtained. Many are introduced not so much to be solved as to be tackled. The value oí a problem is not so much in coming up with the answer as in the ideas and attempted ideas it forces on the would-be solver. Others are included im anticipation of materia! to be developed later, the hope and rationale for this being both to lay the groundyork for the subsequent theory and also to make more natural ideas, definitions, and arguments as they are introduced. Several problems appear more than once. Problems, which for some reason or other seem difficult to me, are often starred (sometimes with two stars). However, even. here there will be no agreement among mathematicians; many will feel that some unstarred probleros should be starred and vice versa. Natuxally, I am indebted to many people for suggestions, comments and eriticisms, To mention just a few of these: Charles Curtis, Marshall Hall, Nathan Jacobson, Arthur Mattuck, and Maxwell Rosenlicht. I owe a great deal to Daniel Gorenstein and Irving Kaplansky for the numerous con- versations we have had about the book, its material and its approach. Above all, I thank George Seligman for the many incisive suggestions and remarks that he has made about the presentation both as to its style and to its content. 1 am also grateful to Francis McNary of the staff of Ginn and Company for his help and cooperation. Finally, I should like to ex- press my thanks to the John Simon Guggenheim Memorial Foundation; this book was in part written with their support while the author was in. Rome as a Guggenheim Fellow. tom n Seg ame ço Ie go Contents 1. Prmiminany NoTroNS « Bet Theory . Mappings . The Integers . 2. Group THpory « Definition of a Group . . Some Examples of Groups . . Some Preliminary Lemmas . Subgroups . À Counting Principle . . Normal Subgroups and Quotient Groups . Homomorphisms . Vo . Automorphisms . Cayley's Theorem . Permutation Groups 1, ia. Another Counting Principle Sylow's Theorem . . 3. Ring THEORY « Definition and Examples of Rings . Some Special Classes of Rings . Homomorphisms . . . Ideals and Quotient Rings . More Ideals and Quatient Rings . The Field of Quatients of an Integral Domain . Euclidean Rings . . - ca . À Particular Euolidean Ring Topics im Algebra CHAPTER 1 Preliminary Notions One of the amazing features of twentieth century mathematics has been. its recognition of the power of the abstract approach. This has given rise to a large body of new results and problems and has, in fact, led us to open up whole new areas of mathematics whose very existence had not even been suspected. In the wake of these developments has come not only a new mathematies but a fresh outlook, and along with this, simple new proofs of difficult clas- sical results. The isclation of a problem into its basic essentials has often revealed for us the proper setting, in the whole scheme of things, of results considered to have been special and apart and has shown us interrelations between areas previously thought to have been unconnected. The algebra which has evolved as an outgrowth of all this is not only a subject with an independent life and vigor—it is one of the important cur- rent research areas in mathematies—but it also serves as the unifying thread which interlaces almost all of mathematics-—geometry, number theory, analysis, topology, and even applied mathematics. This book is intended as an introduetion to that part of mathematics that today goes by the name of abstract algebra. The term “abstract” is a highly subjective one; what is abstract to one person is very often concrete and down-to-earth to another, and vice versa. In relation ta the current research activity in algebra, it could be described as “not too abstract”; from the point of view of someone schooled in the caleulus and svho is seeing the pres- ent material for the first time, it may very well be described as “quite ab- atract.” Be that as it may, we shall concern ourselves with the introduction and development of some of the important algebraic systeme-—groups, rings, vector spaces, fields. An algebraic system. can be described as a set of objects together with some operations for combining them. Priox to studying sets restricted in any way whatever—for instance, with operations—it will be necessary to consider sets in general and some notions about them. At the other end of the spectrum, we shall need some informa- tion about the particular set, the set of integers. It is the purpose of this chap- ter to discuss these and to derive some results about them which we can call upon, as the occasions arise, later in the book. sEc. 1 SET THEORY 3 when we say that x is in A or x is in B we mean 2 is in at least one of A or B, and may be in both. Let us consider a few examples of the union of two sets. For any set 4, AUA=A;im fact, whenever Bisa subsetof 4, 4 U B = A. If 4 isthe set [x1, 12, ta) (Le., the set whose elements are 21, t2, 13) and if B is the set fy, ya, 21), then A U B = fm, xo, xa, 41, 2). TE A is the set of all blonde- haired people and if B is the set of all people who smoke, then 4 U B con- sists of all the people who either have blonde hair or smoke or both. Pic- torially we can illustrate the union of the two sets 4 and B by: Here, 4 is the circle on the left, B that on the right, and 4 U Bisthe shaded part. DerinirIoN. The intersection of the two sets A and B, written as 4 N B, is the set [z|z € 4 and x E B). The intersection of 4 and B is thus the set of all elements which are both in A and in B. In analogy with the examples used to illustrate the union of two sets, let us see what the intersections are in those very examples. For any set 4, ANA = A;in fact, if B is any subset of 4 then 4 N B = BB. KH A is the set (21, 22, 43) and B the set (7h, 42, 21), then AN B= [x] (we are supposing no q is an x). If A is the set of all blonde-haired people and if B is the set of all people that smoke, then 4 N B is the set of all blonde-haired people who smoke. Pictorially we can illustrate the intersec- tion of the two sets 4 and B by: Here 4 is the circle on the left, B that on the right, while their intersection is the shaded part. 4 —PRELIMINARY NOTIONS em, 1 Two sets are said to be disjoint Il their intersection is empty, that is, is the null set. For instance, ií A is the set of pos e integers and B the set of negative integers, then A and B are disjoint. Note however that if C is the set of nonnegative integers and D is the set of nonpositive integers, then they are not disjoint for their intersection consists of the integer 0, so is not empty. Before we generalize union and intersection from two sets to an arbitrary number of them, we should like to prove a little proposition interrelating union and intersection. This is the first of a whole host of such results that esn be proved; some of these can be found in the problems at the end of this section. Proposrriox. For any ihree seis, A, B, C we have ANBUO=(ANBUANO. Proof. The proof will consist of showing, to begin with, the relation (ANBJU(ANCCAN(BUC) and then the converse relation ANBUOQCIANBU(ANO. We first dispose of (ANB)U(ANCCAN(BUC. Because BCBUC, it is immediste that ANBCAN(BUC). In a similar manner, ANCCAN(BUC). Therefore ANBUANOCANEBUQUANBUCG=ANÇUO. Now for the other direction. Given an element zE AN(BUO), first of all it must be an element of 4. Secondly, as an elementin BU Cit is either in B or in €. Suppose the former; then as an element both of À and of B, z mustbein 4 MN B. The second possibility, namely, « € €, leads us to 20 ANC. Thus in either eventuality zC(ANBJU(ANO, whence4 NBUOQCIANBDUANO. The two opposite containing relations combine to give us the equality asserted in the proposition. We continue the discussion of sets to extend the notion of union and of intersection to arbitrary collections of sets. Given a set 7 we say that Y serves as an indez sei for the family 5 = (Aa) of sets if for every « € T there exists a set A, im the family 5. The index set T can be any set, finite or infinite. Very often we use the set of nonnega- tive integers as an index set, but, we repeat, T can be any (nonempty) set. By the union of the sets 4, where aisin 7, we mean the set [2x|t E Aa for at least one x in T). We shall denote it by U Au. By the intersection «Er of the sets Au, where a isin 7, we mean the set Íx|z € As for every a«€ T); we sball denote it by () 4a. The sets Aq are mutually disjoint açr for a 8, Aa N Ag is the null set. For instance, if S is the set of real numbers, and if T is the set of rational 6 PRELIMINARY NOTIONS ca. 1 A subset Rof A X A is said to define an eguivalence relation on A if: (1) (1,0) ERforalge A; (2) (0,5) E R implies 6,9) E R; (3) (a,b) € R and (b,c) E R imply that (g,0) E R. Instead of speaking about subsets of 4 X A we can speak about a binary relation (one between two elements of 4) on À itself, defining b to be re- Jated to a if (a, b) € R. The properties (1), (2), (3) of the subset R inmedi- ately translate into the properties (1), (2), (3) of the definition below. Derrsrriox. The binary relation, —, on 4 is said to be an eguivalence relationon A Eforala, db, cin A: MD ama; (2) e bimpliesb- a; (8) e-bandb-cimplya-e. The first of these properties is called reflexivity, the second, symmetry, and the third, trensitivity. The concept of an equivalence relation is an extremely important one and plays a central role in all of mathematics. We illustrate it with a few examples. Example 1, Let S be any set and define a = b, fora, DES, if and only if a = b, This clearly defines an equivalence relation on S. In fact, an equiva- lence relation is a generalization of equality, measuring equahty up to some property. Example 2. Let S be the set of all integers. Given a, 5 €. 8, define q b fa — bis an even integer, We verily that this defines an equivalence rela- tion of S. (1) Since 0=a—a is even, aq. (2) If ab, thatis, ifa—biseven, thenb—-a=—(a — b) às also even, whence b q. (8) If ab and be, then both o — b and b — c are even, whence a-ec=(a-b)+(b-c) is also even, proving that a = e. Example 3. Let S be the set of 21] integers and let > 1 be a fixed integer. Define fora, DES, a-bifa — bis a multiple of n. We leave it as an exercise to prove that this defines an equivalence relation 0n.8. Example 4. Let S be the set of all triangles in the plane. Two triangles axe defined to be equivalent if they are similar (Le., have corresponding angles equal). This defines an equivalence relation on. 5. sec. 1 SET THEORY 7 Example 5. Let 8 he the set of points in the plane. Two points q and b are defined to be equivalent if they are equidistant from the origin. 4 simple check verifies that this defines an equivalence relation on &, There are many more equivalence relations; we shall encounter a few as we proceed in the book. Derintrron. If 4 is à set and if — is an equivalence relation on 4, then the equivalence class of 4 € A is the set [x E Alex). We write it as ela). In the examples just discussed, what are the equivalence classes? In Example 1, the equivalence class of « consists merely of a itself. In Example 2 the equivalence class of a consists of all the integers of the form a + 2m where m =0, +1, +2,...; in this example there are only two distinct equivalence classes, namely, cl(0) and cl(1). In Example 3, the equivalence class of a consists of all integers of the form a + kn where k = 0, +1, +2,...; here there are n distinct equivalence classes, namely cl(0), et), ..., cl(r — 1), In Example 5, the equivalence class of a consists of all tbe points in the plane which lie on the circle which bas its center at the origin and passes through q. Although we have made quite a few definitions, introduced some con- cepts, and have even established a simple little proposition, one could say in all fairness that up to this point we have not proved any result of real substance. We are now about to prove the first genuine result in the book. The proof of this theorem is not very difficult—actually it is quite easy-— but nonetheless the result it embodies will be of great use to us. Tenorey la. The distinct equivalence classes of an equivalence relation on A provide us with à decomposibon of À as a union of mutualiy disjoint subsets. Conversely, given a decomposition of À as a umen of mulualky dis- joini, nonempty subsets, we can define an equiralence relation on A for which these subseis are the distinct equivalence classes. Proof. Let the equivalence relation on 4 be denoted by —. We first note that since for any a E A,a — a, a must be in ci(a), whence the union of the cl(a)'s is all of 4. We now assert that given two equivalence classes they are either equal or disjoint. For, suppose that cl(a) and cl(b) are not disjoint; then there is an element 3 € el(a) N cl(b). Since x € cl(a), az; since + € cl(b), bx whence by the symmetry of the relation, emb. However, aa and «—b by the transitivity of the relation forces qb. Suppose, now, that y € cl(b); thus by. However, from a band by, we deduce that a = y, that is, that 7 € cl(a). Therefore, every element in cl(b) is in cl(a), which proves that ci(b) E cl(a). The argu- seo. 2 MapPINGS O Prove the following laws that govern these operations: DArBAC=4A+HB+HO. MABHO=AB+A-C. (0) 44 =4. (d) A+ A = null set. (JUAL-B=A4-CthnB=C. (The system just described is an example of & Boolean algebra.) 10, For the given set and relation below determine which define equiva- lence relations. (a) Sis the set of all people in the world today, « = b if a and b have ap ancestor in common. (b) Sis the set of all people in the world today, « — bia lives within 100 miles of b. (c) 8 is the set of all people in the world today, a = bif a and b have the same father. (d) 8 is the set of real numbers, a “bifa = b. (e) S is the set of integers, e — b if both a > b and b > a. (f) S is the set of all straight lines in the plane, a = b if a is parallel tob. 11. (a) Property 2 of an equivalence relation states that if ab then ba; property 3 states that if ab and be then ac. What is wrong with the following proof that properties 2 and 3 imply property 1? Let ab; then ba, whenes, by property 3 (usinga = 0),a- a. (b) Can you suggest an alternative of property 1 which will insure us that properties 2 and 3 do imply property 1? 12. In Example 3 of an equivalence relation given in the text, prove that the relation defined is an equivalence relation and that there are exactly n distinct equivalence classes, namely, cl(0), cl(1), ..., cl(n — 1). 18. Complete the proof of the second half of Theorem 1.a. 2. Mappings. We are about to introduce the concept of a mapping of one set into another. Without exaggeration this is probably the single most important and universal notion that runs through all of mathematics. It is hardly a new thing to any of us, for we have been considering mappings from the very earliest days of our mathematical training. When we were asked to plot the relation y = x? we were simply being asked to study the particular mapping which takes every real number onto its square. Loosely speaking, a mapping from one set, 8, into another, T, is a “rule” (whatever that may mean) that associates with each element in $ a umque element tin 7. We shall define a mapping somewhat more formally and precisely but the purpose of the definition is to allow us to think and speak 10 PRELIMINARY NOTIONS cm, 1 in the above terms. We should think of them as rules or devices or mech- anisms that transport us from one set to another, Let us motivate a little the definition that we will make. The point of view we take is to consider the mapping to be defined by its “graph.” We illustrate this with the familiar example y = «? defined on the real numbers 8 and taking its values also in. S. For this set 8, S X 8, the set of all pairs (a, b) cam be viewed as the plane, the pair (a, b) corresponding; to the point whose coordinates are q and b, respectively. In this plane we single out ali those points whose coordinates are of the form (x, x?) and call this set of points the graph of y = «2. We even represent this set pictorially as To find the “value” of the function or mapping at the point 2 = a we look at the point in the graph whose first coordinate is a and read off the second coordinate as the value of the function at 2 = a. This is, no more or less, the approach we take in the general setting to de- fine a mapping from one set into another. Derixrwon. IS and T are noneropty sets, then a mapping from Sto T isa subset, M, of 8 X T such that for every s E S there is a unique t E T such that the ordered pair (s,t) isin M. This definition serves to make the concept of a mapping precise for us but we shall almost never use it in this form, Instead we do prefer to think of a mapping as a rule srhich associates with any element s in $ some element tin T, the rule being, associate (or map) SE S mthiE Tif and only if (st) € M. We shall say that é is the image of s under the mapping. Now for some notation for these things. Let o be a mappivg from S to T;we often denote this by writing c:S — TorS 5 TH tis the image of s under « we shall sometimes write this as cis — +; more often, we shall represent this fact by é = so. Note that we write the mapping o on the right. There is no over-all consisteney in this usage; many people would write itast = c(s). Às a general rule, algebraists write mappings on the right, other mathemaaticians writing them on the left. In fact, we shall not be absolutely consistent in this ourselves; when we shall want to emphasize the functional nature of o we may very well write é = o(s). 12 PRELIMISARY NOTIONS cm. 1 Go = the subset, S, of 8, ag = Íz1), 4 = fx»). The relation of 8 to S*, in general, is a very interesting one; some of its properties are examined in the problems. Example 9. LetSbeaset, T = St: defimner:S — Tbysr = complement ofisjinS=58-— (5). Example 10. Let 8 be a set with an equivalence relation, and let 7 be the set of equivalence classes in 8 (note that T is a subset of 8*). Define r:5 — T by sr = cl(s). We leave the examples to continue the general discussion. Given a map- ping 7:8 — T we define for i E T, the inverse image of é with respect to 7 to be the set [s € S|t = sr). In Example 8, the inverse image of E is the subset of S consisting of the even integers. It may happen that for some tin T that its inverse image with respect to is empty, that is, t is not the image under 7 of any element in S. In Example 3 which «as discussed the element, (4, 2) is not the image of any element in S under the r used; in Example 9, 8, as an element in 8*, is not the image under the r used of any element in S. Derrinirion. The mapping « of S into T is said to be onto T given tE T there exists an element s € S such that ! = sr. If we call the subset, Sr = [z € T|x = sr for some s € S) the image of S under + then 7 is onto if the image of S under 7 is all of T. Note that in Examples 1, 4, 5, 6, 7, 8, and 10 the mappings used are all onto. Another special type of mapping arises often and is important: the one- to-one mapping. DerrsrrioN. The mapping 7 of S into T is said to be a one-to-one mapping whenever s; = so then syr =* gar. In terms of inverse images, the mapping 7 is one-to-one if for any +€ T the inverse image of t is either empty or is a set consisting of one element. Tn the examples discussed, the mappings in Examples 1, 3, 7, and 9 are all one-to-one. When should we say that two mappings from S to T are equal? A natural definition for this is that they should have the same effect on every element; of S, that is, the image of any element in. S under each of these mappings should be the same. In a little more formal manner: Derrxiriox. The two mappings o and + of S into T are said to be equal ifso=serforeverysEs. Consider the following situation: We have a mapping o from Sto T and another mapping 7 from 7 to U. Can we compound these mappings to sec. 2 MAPPINOS 13 produce a mapping from S to U? The most natural and obvious way of doing this is to send a given element s, in S, in two stages into U, first by applying o to s and then applying 7 to the resulting element se in T. This is the basis of the Derinrrion. If o:S — Tandr:T — U then the composition of o and (also called their product) is the mapping o o r:S — U defined by means of s(co+) = (so)rforevery s E 8. Note that the order of events reads from leít to right; o o 7 reads, first perform o and then follow it up with 7. Here, too, the left-right business is not a uniform one. Mathematicians who write their mappings on the left «rould read o o 7 to mean first perform 7 and then 9. Accordingty, in reading a given book in mathematics one must make absolutely sure as to what convention is being folowed in writing the product of two mappings. We reiterate, for us o o 7 will ahvays mean: first apply o and then r. We illustrate the composition of o and + with a few examples. Example 1. Let 8 = (x, xo, 73! and let T = 8. LetoiS — S be defined byrxoc=2a, go=z to=madrS>sS byar=ar, marta, gar = x2. Thuszy(007) = (mo)r = ator = ta, da(o 07) = (tao)7 = tar = 29, galo or) = (tgo)r = mir = x. At the same time we can compute 7 o q, because in this case it also makes sense. Now zy(700) = (zyno= (x0) = to, aa(r00) = (uao)o = aa = 21, to(r 00) = (agro = mos = dg. Note that xo = z/(r o 0), wheteas xy = zi(c e 7) whence cor rod, Example 2. Let S be the set of integers, T the set S X'S, and suppose ciS > P is defmed by mo = (m — 1,1). Let U = S and suppose that qiT7— Ul=S) is defined by (mnr=m+n Thus coriS —S whereas reciT —s T; even to speak about the equality of cor and + o 9 would make no sense since they do not act on the same space. We now compute o o as a mapping of ;S into itself and then 7 oc as one on Tinto itself, Givenm E S,mo = (m— 1, 1) vwhence m(c o 7) = (mo)r = (m — 1, Ir =(m-1+1=m. Thuscoer is the identity mapping of S into itself. What about voc? Given GM) ET, (mn =m+n whereby tm, nro = (Cm ndo =(m+ns=(m+n-— al, 1). Note that wo o is not the identity map of T into itself; it is not even an onto mapping of 7, Example 8. Let S be the get of real numbers, 7 the set of integers, and U = (E,0). Define ciS — T by so = largest integer less than or equal tos, andriT — U defined by nr = E if niseven, nz = 0 ifnisodd. Note that in this case voo cannot be defined. We compute o o 7 for two real 
