A course in ABStract algebra

NICHOLAS JACKSON A COURSE IN ABSTRACT ALGEBRA D R A F T: JA N UA RY , To Abigail and Emilie We may always depend upon it that algebra which cannot be translated into good English and sound common sense is bad algebra — William Kingdon Clifford (1845–1879), The Common Sense of the Exact Sciences (1886) 21 Preface Mathematics is written for mathematicians — Nicolaus Copernicus (1473–1543), preface to De Revolutionibus Orbium Cœlestium (1543) Contents Preface v Groups 1.1 Numbers 1.2 Matrices 17 1.3 Symmetries 20 1.4 Permutations 24 Subgroups 39 2.1 Groups within groups 39 2.2 Cosets and Lagrange’s Theorem 49 2.3 Euler’s Theorem and Fermat’s Little Theorem 62 Normal subgroups 69 3.1 Cosets and conjugacy classes 69 3.2 Quotient groups 80 3.A Simple groups 91 Homomorphisms 103 4.1 Structure-preserving maps 103 4.2 Kernels and images 110 4.3 The Isomorphism Theorems 116 Presentations 133 5.1 Free groups 134 5.2 Generators and relations 144 5.3 Finitely generated abelian groups 165 5.A Coset enumeration 178 5.B Transversals 186 5.C Triangles, braids and reflections 197 viii a course in abstract algebra Actions 205 6.1 Symmetries and transformations 205 6.2 Orbits and stabilisers 213 6.3 Counting 220 Finite groups 233 7.1 Sylow’s Theorems 233 7.2 Series of subgroups 244 7.3 Soluble and nilpotent groups 255 7.4 Semidirect products 272 7.5 Extensions 277 7.A Classification of small finite groups 291 Rings 317 8.1 Numbers 317 8.2 Matrices 326 8.3 Polynomials 328 8.4 Fields 332 8.A Modules and representations 338 Ideals 343 9.1 Subrings 343 9.2 Homomorphisms and ideals 351 9.3 Quotient rings 359 9.4 Prime ideals and maximal ideals 368 10 Domains 377 10.1 Euclidean domains 377 10.2 Divisors, primes and irreducible elements 385 10.3 Principal ideal domains 391 10.4 Unique factorisation domains 395 10.AQuadratic integer rings 408 11 Polynomials 421 11.1 Irreducible polynomials 421 11.2 Field extensions 425 11.3 Finite fields 439 11.4 Field automorphisms and the Galois group 442 11.5 The Galois Correspondence 447 ix A 11.6 Solving equations by radicals 464 11.AGeometric constructions 474 Background 481 A.1 Sets 481 A.2 Functions 487 A.3 Counting and infinity 492 A.4 Relations 496 A.5 Number theory 499 A.6 Real analysis 502 A.7 Linear algebra 503 Index 505 492 a course in abstract algebra infinite sets The key observation is that a bijection defines an exact correspondence between its domain and codomain We can use this as a way of counting the elements of a finite set: we construct a bijection between our chosen set and a subset of natural numbers of the form {1, 2, , n} for some n ∈ N For example, taking the set A = { a, b, c, d} we can define a bijection f : A → B where B = {1, 2, 3, 4} In fact, we can define 4! = 24 different such bijections, of which f ( a) = 1, f (b) = 3, f (c) = 4, f (d) = is one If B has any fewer elements then no function f : A → B can be injective, and if it has any more elements then no function f : A → B can be surjective So the only case in which a bijection can exist is when A and B have the same number of elements And since we know that B has elements, A must too O God, I could be bounded in a nutshell, and count myself a king of infinite space – were it not that I have bad dreams — William Shakespeare (1564–1616), Hamlet II:2 (c.1600) A.3 Counting and infinity We can extend this approach to infinite sets as well Suppose that we take some infinite set A and manage to construct a bijection f : A → N Then by the discussion above these two sets must have the same “number” of elements, although it’s usually unwise to treat infinity as a conventional number because it behaves in particularly counterintuitive ways In fact, as we’ll see in a little while, there isn’t even just one infinity, but an entire hierarchy of them Our first counterintuitive consequence of this discussion is that there are as many natural numbers as there are even natural numbers: Proposition A.23 There is a bijection f : 2N → N defined by f (n) = n Proof Every even natural number is of the form n = 2k for some k ∈ N, so the function f is well-defined in the sense that every element of 2N maps to an element of N The function f is injective, since if f (n) = f (m) we have 12 n = 12 m and hence n = m It is also surjective, since for any m ∈ N we can find 2m ∈ 2N such that f (2m) = 12 (2m) = m Therefore f is a bijection Equally surprisingly, there are as many integers as natural numbers: Proposition A.24 There is a bijection f : Z → N defined by  2n + if n 0, f (n) = −2n if n < background 493 Proof The function f is well-defined: it maps every integer to a natural number It is injective, since it maps distinct negative integers to distinct even natural numbers and non-negative integers to distinct odd natural numbers And it is surjective since for every odd natural number m we can find an integer n = 21 (m − 1) with f (n) = m, and for every even natural number m we can find a negative integer n = − 12 n such that f (n) = m Therefore f is a bijection It’s useful to have a specific word to describe sets for which there exists a bijective function to the natural numbers N Since the natural numbers are the “counting numbers” that we meet quite early in primary school, a bijection to N is really just a way of counting the elements of our chosen set Wikimedia Commons Definition A.25 Let A be a set, and suppose either that A is finite, or Georg Ferdinand Ludwig Philipp that there exists a bijection f : A → N Then we say A is countable Cantor (1845–1918) was born in St PeIf A is finite, then the cardinality or order | A| of A is the number of elements in A If A is infinite and countable then we say it has cardinality | A| = ℵ0 (pronounced aleph null or aleph zero) We will often use the leminiscate or analemma symbol ∞ to denote infinity in general, but ℵ0 specifically denotes the countable infinity, the cardinality of the natural numbers N Even more surprisingly, it turns out that the rational numbers Q are countable The proof of this celebrated result is due to the German mathematician Georg Cantor (1845–1918) Proposition A.26 The set Q of rational numbers is countable Proof We start by proving that the positive rationals Q+ are countable To this, we display them in a grid as follows: 1 ··· 2 ··· 3 3 ··· 4 4 ··· 5 5 5 ··· 6 6 6 ··· tersburg but in 1860, the family moved to Frankfurt Here Georg, already an accomplished violinist, excelled in mathematics and subsequently attended the Universities of Zürich, Berlin and Göttingen After completing a doctoral dissertation in number theory at Berlin in 1867, he moved to the University of Halle and was promoted to Professor in 1879 Much of his most important work was achieved between 1874 and 1884, a period overshadowed by bitter disputes with the influential German mathematician Leopold Kronecker (1823–1891), a founder of the constructivist school in mathematics, who had fundamental objections to Cantor’s work Kronecker, head of department at Berlin, also blocked Cantor’s attempts to secure an academic post there The stress of this situation took its toll, and Cantor suffered the first of several nervous breakdowns, spending time in a sanatorium with severe depression He moved away from mathematics for a time, focusing instead on philosophy and English literature He researched the authorship of the plays of William Shakespeare (1564–1616), and argued in favour of Francis Bacon (1561–1626) He suffered from depression for most of the rest of his life, a condition that was exacerbated by the death of his youngest son Rudolph in 1899 and occasionally fierce criticism of his work on transfinite set theory He died in a sanatorium in 1918, roughly two months before his 73rd birthday 494 a course in abstract algebra Following the arrows yields the list 1 1 1, 1, 2, 3, 2, 1, 1, 2, 3, 4, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, However, each rational number will occur infinitely many times (for example, 12 = 24 = 36 = · · · ) so we must delete from this list the second and subsequent occurrences of every given rational number This gives the list 1 1 1, 1, 2, 3, 1, 1, 2, 3, 4, 5, 1, 1, 2, 3, 4, 5, 6, which contains every single positive rational number exactly once We can then construct a bijection f : Q+ → N by mapping each rational number to the positive integer given by its position in the list So, for example, 25 → 16 and so forth This function, while not constructed by means of a neat formula, is nevertheless well-defined and bijective, and so |Q+ | = |N| = ℵ0 as claimed We can extend this to provide a bijection g : Q → N by using a similar trick to the one used in Proposition A.24: we define   2 f (q) + if q > 0,  g(q) = if q = 0,    f (−q) if q < This function g is a bijection, and thus Q is countable The obvious, naïve approach to constructing a bijection f : Q → N is to start in the top left-hand corner of the grid and proceed horizontally, row by row, accounting for all the rational numbers The problem is that there are (countably) infinitely many rational numbers on the first row, so we never actually get as far as the second or subsequent rows Cantor’s method avoids this problem by traversing the grid in a zig-zag pattern that doesn’t miss anything out: eventually we’ll reach any given rational number in a way that doesn’t happen if we follow the obvious row-by-row method What about the next set in our standard hierarchy of number systems? Are the real numbers countable? There are certainly infinitely many of them, because R contains Q as a proper subset, but it turns out we can’t construct a bijection f : R → N no matter how carefully we arrange them The following result is also due to Cantor, and has become known as Cantor’s diagonal argument Proposition A.27 The set R of real numbers is not countable The following proof uses a technique called proof by contradiction or reductio ad absurdum, which we’ll discuss in more detail later The background central idea is that we assume the opposite of what we’re trying to prove, and then show that the logical consequences of that are so obviously incorrect or unthinkable that the original proposition must have been true all along Proof First we show that the interval (0, 1) = { x : < x < 1} is uncountable Suppose for the moment that (0, 1) is countable after all That means that there is a bijection f : (0, 1) → N, and hence we can (in principle, if not in practice) write down an infinite list of all the real numbers in the open interval (0, 1) This yields a sequence x = ( x1 , x2 , ) where xi is the unique real number satisfying f ( xi ) = i ∈ N We can write every number y ∈ (0, 1) as an infinite decimal expansion of the form y = 0.y1 y2 y3 where each digit yi ∈ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} To avoid ambiguity we write any decimal expansion of the form 0.y1 y2 yk 9999 (where yk = 9) as 0.y1 y2 (yk +1)0000 instead, and we pad any finite-length decimal expansion to an infinite one by appending an infinite sequence of trailing zeros after the last nonzero digit Our sequence x of all real numbers from (0, 1) can be written out as infinite decimal expansions as follows: x1 = x11 x12 x13 x14 x2 = x21 x22 x23 x24 x3 = x31 x32 x33 x34 x4 = x41 x42 x43 x44 We now construct a new number x0 by taking the diagonal digits of this list x1 = x11 x12 x13 x14 x2 = x21 x22 x23 x24 x3 = x31 x32 x33 x34 x4 = x41 x42 x43 x44 to get 0.x11 x22 x33 x44 495 496 a course in abstract algebra and then adding to each digit using modulo–10 arithmetic This yields x0 = 0.( x11 +1)( x22 +1)( x33 +1)( x44 +1) , a perfectly valid real number that lies in the interval (0, 1) but doesn’t appear anywhere in the existing sequence x The reason for this is that for any n ∈ N, the number xn differs from x0 at the nth digit Therefore the sequence x didn’t contain all of (0, 1), and in fact no such sequence can exist, so (0, 1) is not countable after all Since R contains (0, 1) as a proper subset, R isn’t countable either This proves that |R| = ℵ0 , and since N ⊂ R we already know that |N| |R, so it must follow that |N| < |R| and hence we need a new infinite cardinal We say that R is uncountable, and has cardinality ℵ1 (pronounced aleph one) It is possible to construct hierarchies of successively larger infinite cardinals, but we won’t go into the details For a very readable discussion of infinity and related matters see the book Infinity and the Mind by the mathematician, logician, artist and science fiction writer Rudy Rucker.4 The Argentinian magic realist writer Jorge Luis R Rucker, Infinity and the Mind, Princeton University Press (2005) Borges (1899–1986) explores ideas of infinity in several of his stories, J L Borges, The Library of Babel, in: in particular The Library of Babel, The Aleph and The Book of Sand Labyrinths, Penguin (1970) 78–86 Bijective functions also enable us to properly answer the question about the commutativity and associativity of cartesian products As noted earlier, in the discussion following Definition A.7, in general J L Borges, The Book of Sand, in: The A × B = B × A for arbitrary sets A and B There is, however, an obvious Book of Sand and Shakespeare’s Memory, bijection f : A × B → B × A given by f ( a, b ) = (b, a ) for all a ∈ A and Penguin (2001) 89–93 b ∈ B J L Borges, The Aleph, in: The Aleph and Other Stories, Penguin (2000) 118– 133 Similarly, although A×( B×C ) = ( A× B)×C, we can define a bijection g : A×( B×C ) → ( A× B)×C by g(( a, b), c) = ( a, (b, c) and therefore it makes sense in almost all cases to regard these as effectively the same as each other and the threefold Cartesian product A× B×C A.4 Relations Earlier, we noted that a set has, by default, no particular predetermined structure Most of the rest of this book is concerned with the study of some particularly important structures that arise naturally in various contexts Now we will examine another class of structures that can be defined on sets We often find it necessary or useful to compare individual elements background of sets in various ways For example, sometimes we need to know in what circumstances two arbitrary elements are equal to each other Or we might be interested in whether one element is greater than another (if the set in question actually has a coherent concept of “greater than” or “less than”) More generally, we sometimes find it necessary to study other connections between elements In number theory, for example, we are often interested in whether some integer is a factor of another We’d like to formalise this idea and see what important properties arise To start with, any two elements of a set are either related in a specific way or they aren’t So what we want is a way of assigning a value of either “true” or “false” to a given ordered pair of elements, depending on whether those elements are related in the specified way or not One way of looking at this is as a function f : A× A → {true, false} Alternatively, we can define a relation by listing the ordered pairs of related elements This approach views a relation as a subset of the cartesian product A × A Both approaches are valid and equivalent, and each has its own advantages, so we present both Definition A.28 A (binary) relation on a set A is a function f : A × A → {true, false}, and we say that two given elements a, b ∈ A are related if f ( a, b) = true Equivalently, a (binary) relation on a set A is a subset B ⊆ A × A, and we say that two given elements a, b ∈ A are related if ( a, b) ∈ B The notation f ( a, b) = true or ( a, b) ∈ B is rather cumbersome, and obscures the comparative aspect of the relation under discussion, so in most cases we will represent a given relation with a symbol such as ∼ placed between the two elements This enables us to write more intuitive expressions such as a ∼ b if a is related to b, and a ∼ b if not Example A.29 For any set A we can define the equality relation = defined by  true if a = b, f ( a, b) = false if a = b Example A.30 Let f : N × N → {true, false} be defined by  true if m is a factor of n, f (m, n) = false otherwise This determines the relation | or “divides”: we say that m|n if and only if f (m, n) = true Another important relation is the notion of an ordering on a set: 497 498 a course in abstract algebra Example A.31 Suppose A ⊆ R Then the “less than” relation < is given by  true if a < b, f ( a, b) = false if a b We can define the “less than or equal” relation in a similar way Each of these relations have subtly different properties Is it always the case that a given element is related to itself? In the case of = it so happens that a = a for any element a ∈ A, but it’s not the case that a < a in general (or indeed at all) for subsets A ⊆ R, although clearly a a in all cases Furthermore, n|n for any n ∈ N Does the order of the elements matter? If a ∼ b then is it necessarily true that b ∼ a? If a = b then clearly b = a in all cases But if a < b it’s certainly not the case that b < a, and if m|n it’s not true that n|m (except when m = n) What about comparisons of three arbitrary elements? If a = b and b = c then clearly a = c Also, if a < b and b < c then it’s necessarily the case that a < c And if p|q and q|r then p|r Here, rock beats scissors (because the rock blunts the scissors), scissors beat paper (because the scissors cut the paper) and paper beats rock (because the paper wraps the rock) Every other outcome is a draw This game can be generalised to any odd number of players: there is a popular five-player variant called “rock– paper–scissors–lizard–Spock” Are there any relations that don’t satisfy this third property? Probably the best known example is the old game of “rock–paper–scissors” (sometimes called roshambo or ick ack ock).8 We can define a relation “beats” on the set {rock, paper, scissors} and then observe that although rock beats scissors, and scissors beat paper, rock doesn’t beat paper This relation also doesn’t satisfy the other two conditions Definition A.32 A relation ∼ defined on some set A is said to be (i) (ii) (iii) reflexive if a ∼ a for all a ∈ A, symmetric if a ∼ b implies that b ∼ a for all a, b ∈ A, and transitive if a ∼ b and b ∼ c together imply that a ∼ c for all a, b, c ∈ A So, in this terminology, = is reflexive, symmetric and transitive; < is transitive only; and and | are both reflexive and transitive but not symmetric The relation “beats” in rock–paper–scissors is neither reflexive, symmetric nor transitive A relation which satisfies all three of these conditions is in some sense stronger or more exclusive than one that satisfies only some or none of them We can view such a relation as a kind of generalised notion of equality or equivalence: Definition A.33 A relation ∼ on a set A is said to be an equivalence relation if it is symmetric, reflexive and transitive The motivating example of an equivalence relation is, of course, the background equality relation =, but there are others that arise naturally in certain contexts In Section 2.2 we formulate a particularly important class of equivalence relations on groups, and use it to prove Lagrange’s Theorem.9 Probably the best-known example of an equivalence relation, apart from = itself, is that of congruence modulo n 499 Theorem 2.30, page 54 Definition A.34 Two integers a, b ∈ Z are congruent modulo n for some positive integer n ∈ N if ( a−b)|n Or, equivalently, if they both have the same remainder when divided by n We write a ≡n b or a ≡ b (mod n) This is an equivalence relation on the set of integers A.5 Number theory Much of abstract algebra is concerned with extending and generalising facts about the integers to wider classes of objects, such as polynomials, symmetry operations, matrices and so forth In this section we review a few elementary number-theoretic results used elsewhere in the book First we introduce a couple of basic definitions God created the integers; all else is the work of man — Leopold Kronecker (1823–1891), lecture at Göttingen (1886), quoted by Heinrich Weber (1842–1913), Jahresbericht der Deutschen Mathematiker-Vereinigung (1892) 19 Definition A.35 Let a, b ∈ Z Then a is a factor or divisor of, or divides b, if b = ka for some integer k ∈ Z We write a|b if this is so, and a |b if not The greatest common divisor or highest common factor of two positive integers m, n ∈ N is the largest positive integer d ∈ N that divides both m and n We write this as gcd(m, n) If gcd(m, n) = then m and n are said to be coprime or relatively prime The ancient Greek mathematician Euclid of Alexandria (fl 300BC) gives an algorithm for finding the greatest common divisor of two positive integers It appears as Propositions and in Book VII (and 10 This proof makes use of the method later, in a slightly different form, as Propositions and in Book X) of of Proof by Induction, which relies on his celebrated treatise Στοιχεια (Elements), although it almost certainly (indeed, its validity is formally equivalent to) the Well-Ordering Principle: dates back even earlier than that In order to prove the validity of this algorithm, we need a basic fact about natural numbers, namely that the process of division with remainders, which most of us learn at primary school, actually works.10 Theorem A.37 (The Division Theorem) Let a, b ∈ N Then there exist unique integers q, r ∈ Z such that a = bq + r and r < b Proof First, we prove the existence of q and r If a = b then obviously a = 1b + 0, so q = and r = suffice Axiom A.36 (The Well-Ordering Principle) Let S ⊆ N such that S = ∅ That is, let S be a nonempty set of natural numbers Then S has a least element Depending on exactly which version of axiomatic set theory one is working with, this statement is either a basic axiom or a provable proposition We will cheerfully ignore such concerns in the rest of this book 500 a course in abstract algebra If a =b then we proceed by induction on a For a=1 (and thereby b>1) we have a = = 0b + with q = and r = < b Now suppose that a = qb + r for some q, r ∈ Z with adding to both sides of this equation we obtain r < b Then a + = qb + r + Since r < b it follows that r + b If r + = b then a + = (q + 1)b + 0, and if r + < b we have Wikimedia Commons / Raphael (1483–1520), detail from The School of Athens (1509–1510) Almost nothing is known of the life of the ancient Greek mathematician Euclid of Alexandria (fl 300BC), the few historical references having been written centuries later He is believed to have lived during the reign of Ptolemy I of Egypt (c.367–c.283BC), and there are six surviving works that are credibly attributed to him, the best-known of which is the Elements (Στοιχεια) This work, consisting of thirteen books of definitions, axioms and propositions on plane geometry, spatial geometry and number theory, is perhaps one of the most influential mathematical texts of all time: for several centuries it was part of the standard Western university and school curriculum One of the earliest surviving fragments is contained in the Oxyrhinchus Papyrus, dated to around 100AD, but probably the earliest surviving complete texts are an edition prepared in the fourth century AD by Theon of Alexandria (c.335–c.405AD) and a Byzantine palimpsest dating from about 900AD discovered in 1808 by the French historian Franỗois Peyrard (c.17601822) a + = qb + (r + 1) In each case a+1 can be expressed in the desired way, and hence it follows by induction that any a ∈ N can be decomposed in this way too To show uniqueness, suppose that there exist two other integers s, t ∈ Z such that a = sb + t, with t < b Then sb + t = qb + r, which implies that (q − s)b = t − r This means that (t − r ) is divisible by b Since that |t − r | < b, which means that −(b − 1) t−r r, t < b it follows b − The only integer in this range divisible by b is 0, so t − r = and hence t = r Now we have qb + r = sb + t = sb + r, and hence qb = sb, from which we find that q = s Hence q and r are unique We can now state Euclid’s Algorithm, and prove that it works Theorem A.38 (Euclid’s Algorithm) Let a, b ∈ N Then there exist unique integers q1 , , qn+1 (the quotients) and b>r1 > · · · >rn+1 =0 (the remainders) such that a = q1 b + r1 , b = q2 r1 + r2 , r1 = q3 r2 + r3 , r n −2 = q n r n −1 + r n , r n −1 = q n +1 r n + r n +1 , Wikimedia Commons / Oxyrhynchus Papyrus, fragment 29 and gcd( a, b) = rn , the last nonzero remainder background Proof First we apply the Division Theorem (Theorem A.37) to the pair a, b to find the unique integers q1 , r1 satisfying a = q1 b + r1 and r1 < b If r1 = then we stop Otherwise, we apply the Division Theorem again to the pair b, r1 to find the unique integers q2 , r2 such that b = q2 r1 + r2 If r2 = then we stop, otherwise we apply the Division Theorem yet again to the pair r1 , r2 Continuing this process, we obtain a strictly decreasing sequence of remainders r1 > r2 > · · · which are all less than b and strictly non-negative Hence, after finitely many steps, we reach rn+1 = The last step is therefore r n −1 = q n +1 r n + and so rn−1 = qn+1 rn , which means that rn |rn−1 The penultimate step, similarly, is r n −2 = q n r n −1 + r n = q n q n +1 r n + r n = ( q n q n +1 + ) r n and so rn |rn−2 Continuing backwards, we find that rn divides all of the other remainders rn−1 , , r1 Furthermore, since rn |r2 and rn |r1 , it follows that rn also divides b = q2 r1 + r2 And since rn |b, it also follows that rn divides a = q1 b + r1 Thus rn is a factor of both a and b All that remains is to show that rn is the largest such integer; that is, rn = gcd( a, b) Suppose d ∈ N is another common factor of a and b Rearranging the equations obtained via the Division Theorem, we get r1 = a − q1 b, r2 = b − q2 r1 , r3 = r1 − q3 r2 , r n = r n −2 − q n r n −1 Now suppose d| a and d|b Then from the first equation we find that d|r1 , from the second that d|r2 , from the third that d|r3 , and so on Finally, we find that d|rn−2 and d|rn−1 , so d|rn Therefore d rn , and hence rn is the highest such factor That is, rn = gcd( a, b) as claimed Corollary A.39 Let a, b ∈ N, and let d = gcd( a, b) Then there exist integers s, t ∈ Z such that d = sa + tb Proof From the latter part of the above proof, we know that Euclid’s Algorithm ensures the existence of integers q1 , , qn+1 and b > r1 > · · · > rn+1 > such that r1 = a − q1 b, r2 = b − q2 r1 , 501 502 a course in abstract algebra r3 = r1 − q3 r2 , r n −1 = r n −3 − q n −1 r n −2 , d = r n = r n −2 − q n r n −1 Substituting the penultimate expression into the last one, we obtain d = r n = r n −2 − q n ( r n −3 − q n −1 r n −2 ) = (1 + q n q n −1 ) r n −2 − q n r n −3 Hence d = rn can be expressed as a Z–linear combination of rn−3 and rn−2 Now we use the antepenultimate expression on the list to substitute for rn−2 and thereby express d as a Z–linear combination of rn−4 and rn−3 Continuing this process, we eventually obtain an expression for d as a Z–linear combination of a and b, as required The following important fact about coprime integers is easy to prove, and will be used elsewhere in the book Proposition A.40 Let a and b be coprime integers Then if a divides bq then a must divide q Proof If a and b are relatively prime, then by Corollary A.39 there exist integers s and t such that sa + tb = Multiplying both sides of this equation by q we get saq + tbq = q Since a obviously divides saq, and by hypothesis a also divides bq, the entire left hand side of this equation is divisible by a Hence a must also divide the right hand side as well, which means that a divides q as claimed A.6 Real analysis Theorem A.41 (The Intermediate Value Theorem) Let f : R → R be continuous on the closed interval [ a, b] for some a, b ∈ R Then if f ( a) and f (b) have opposite signs, there exists some c ∈ ( a, b) for which f (c) = Theorem A.42 (Rolle’s Theorem) Let f : R → R be continous on the closed interval [ a, b] and differentiable on the open interval ( a, b), and suppose that f ( a) = f (b) Then there exists some c ∈ ( a, b) for which f (c) = background A.7 Linear algebra 503 Index Abel, Niels Henrik, Araucaria, see Graham, John The Art of War, 297 ¯ Aryabhat a, 297 ¯ ¯ Aryabhat ı ya, 297 Bernoulli, Johann, 11 binary operation, Brahmagupta, Cardano, Girolamo, commutative, Darboux, Jean Gaston, 20 Diophantus, Engel, Friedrich, 20 Euler, Leonhard, 11 totient function, 15 Feynman, Richard, Fibonacci, 5, 6, 297 sequence, Galois, Évariste, Graham, John, 27 Jiuzhang suanshu, see Nine Chapters on the Mathematical Art Jordan, Camille, 20 Klein, Felix, 17, 20 Leonardo of Pisa, see Fibonacci Liber Abaci, 6, 297 Lie, Marius Sophus, 20 magma, Mah¯avira, Mathematical Classic of Sun Tzu, 297 Maugham, W Somerset, 303 Nine Chapters on the Mathematical Art, Pauli, Wolfgang, 21 Plücker, Julius, 17 Poincaré, Henri, 22 Sun Tzu, 297 Sun Tzu Suan Ching, see Mathematical Classic of Sun Tzu Young, Grace Chisholm, 17 506 a course in abstract algebra ... calculating sums of three or more natural numbers Formulating addition as a function f ( a, b) in this way, we gain certain formal advantages, but we lose the valuable intuitive advantage that... we say S is a commutative or abelian monoid Monoids also yield a rich field of mathematical study, and in particular are relevant in the study of automata and formal languages in theoretical computer... them in a financial context as a loss or debit Meanwhile, Wikimedia Commons the Indian mathematicians Brahmagupta (598–668) and Mah¯avira (9th Figure 1.1: Page from Nine Chapters on the Mathematical