MATH 351: INTRODUCTION TO ABSTRACT ALGEBRA SUMMER 2017, RUTGERS UNIVERSITY KELLY SPENDLOVE On Algebra Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine. – Sir Michael Atiyah1 ‘Abstract Algebra’ is a study of structure and ‘arithmetic systems’, e.g. groups, rings, fields. Algebra grew out of arithmetic - abstract algebra will axiomatize basic concepts you’ve studied before (in Z, Zn) and we will study their structure. Here is a prototype of a structure/decomposition theorem (we will see this again at the end of Ch. 1) Theorem 0.1 (Fundamental Theorem of Arithmetic). Let n 2 Z with n 6= 0, ±1. Then n is a product of primes, i.e. n = p1p2 ··· pn for pi prime and this factorization is unique up to reordering The key to this class will be working through as many problems as possible. It is crucial to read the book. You should understand every proof in the book. For the most part these notes will follow Abstract Algebra, An Introduction [4], T. Hunger- ford (3rd edition), colloquially known as ‘Baby Hungerford’. (‘Hungerford’ typically refers to his book Algebra in the Graduate Text in Mathematics series). Baby Hungerford (B.H.) is sometimes considered ‘wordy’, but all of those words assem- ble into a book that is cogent. These notes will likely not be as cogent and may suffer under various idiosyncrasies. However, they will draw inspiration from other classic algebra texts such as Hungerford, Dummit and Foote, Jacobson’s Basic Algebra I and M. Artin’s Algebra. Notable about B.H. is that it is unorthodox in pedagogy, viz. introducing rings before groups. These notes will follow this precedent, though perhaps of interest to note is that the classical treatment is groups ! rings ! fields (e.g. Hungerford’s GTM Algebra). 1Mathematics in the 20th Century. Bulletin of the London Mathematical Society. 2002. Recommended reading (15 pages). 1 2 KELLY SPENDLOVE 1.Chapter 1:Arithmetic in Z . for a student the content of a mathematical theory is never larger than the set of examples that are thoroughly understood. – Vladimir Arnol’d2 Two good models for many ideas of abstract algebra are the integers Z = f0, ±1, ±2, :::g and ‘the integers modulo n’ denoted Zn Remark 1.1. What is a ‘model’? We can see this word all over algebra textbooks. This is a loaded term. What is meant by ‘model’ is a specific instantiation of an idea - one that you keep sitting around in your head. When you learn new/abstract concepts about groups, rings, etc, you apply them to the model. That is a good way to get an intuition for how the new/abstract concept works. Page ‘xvi’ of B.H. has a thematic table of contents. The topics of this course are primes and factorization/decomposition, congruence, quotients and structure. We will study these first in Z and Zn and then abstract to rings and groups. Chapters 1 and 2 of our book are mostly a review of these concepts. It is important to work with the con- crete examples of these sections in order to develop a good intuition for the upcoming abstraction. Remark 1.2. What does it mean ‘to abstract’? It is often the case that instances of ideas are first found in applications or are used implicitly. The idea is then recognized as important or interesting in its own right. Then an ‘abstract definition’ is made for the idea. This ‘abstract definition’ should capture and isolate what is unique/characteristic of the idea. Z, Zn work as concrete models for rings and groups. The archetypical example of a ring are the polynomials. For groups: symmetries of a shape3. A conceptual outline to keep in mind for the course may be as follows: Group/ring properties of Z, Zn Rings q /- Groups Ex: Polynomials Ex: Symmetries * Structure and Isomorphism Theorems t 1.1. Division Algorithm. Proposition 1.3 (Well-Ordering Principle). Every nonempty subset of the set of nonnegative integers contains a smallest element. This is a set theoretic axiom.4 This will be the workhorse of many of the proofs to come in this chapter. 2Arnol’d, V. I. (2013). Lectures on partial differential equations. Springer Science & Business Media. 3See S. Strogatz’s article Group Think: https://opinionator.blogs.nytimes.com/2010/05/02/ group-think/. Strogatz is a math professor at Cornell, and one of the best mathematical expositors. 4See https://en.wikipedia.org/wiki/Well-ordering principle MATH 351: INTRODUCTION TO ABSTRACT ALGEBRA SUMMER 2017, RUTGERS UNIVERSITY 3 Theorem 1.4 (Division Algorithm). Let a, b 2 Z be integers with b > 0. Then there exists unique integers q and r such that a = bq + r and 0 6 r < b Remark 1.5. We will apply the well-ordering principle. To set this up, we need a set of nonnega- tive integers. This proof is an archetypical application of the well-ordering principle!! This proof technique will be used over and over again! Proof. Let a, b 2 Z be integers with b > 0. Let S = fa - bx : x 2 Z and a - bx > 0g Step 1) Show that S is nonempty. We must find a value for x such that a - bx > 0. What x can we choose? How about x = -jaj? Then check a + bjaj > 0 Step 2) Find q, r such that a = bq + r and r > 0. By the well-ordering principle S must contain a smallest element, call it r. Then r = a - bq for some q 2 Z. Step 3) Show that 0 6 r < b. We have that 0 6 r by defn of S. We will use a proof by contradiction to show that r < b. Suppose that r > b. Then 0 6 r - b = (a - bq)- b = a - b(q + 1) The right hand side is an element of S. Since b > 0 we have a - b(q + 1) < a - bq = r. This contradicts our choice of r. Step 4) Show that r, q are unique. Suppose a = bq1 + r1 a = bq2 + r2 with 0 6 r1 < b 0 6 r2 < b We can multiply 0 6 r1 < b by -1 to get -b < -r1 6 0. Adding these two equations we have -b < r2 - r1 < b. We have bq1 + r1 = a = bq2 + r2. Rearranging, we have r2 - r1 = b(q1 - q2) Therefore -b < b(q1 - q2) < b. Implying -1 < q1 - q2 < 1. Since q1 - q2 is an integer, it is forced to be zero. Remark 1.6. Is it important that b > 0 as a hypothesis of the theorem? See Exercise 11 in 1.1. Remark 1.7. Why is 0 6 r < b important? If we do not specify this, then q, r need not be unique! Remark 1.8. The division algorithm says something about the structure of Z. This says some- thing about how a breaks apart in terms of b: a is a multiple of b plus some remainder 1.2. Divisibility. An important case of division occurs when for a = bq + r we have remainder r = 0. Let a, b 2 Z with b 6= 0. If a = bc for some c 2 Z then b is said to divide a, written as bja. If b does not divide a, then we write b - a. An important concept concerning divisors is the following: 4 KELLY SPENDLOVE Definition 1.9 (Greatest Common Divisor). Let a and b be integers, not both 0. d is greatest common divisor (gcd) of a and b if (1) dja and djb (2) if cja and cjb then c 6 d Denoted (a, b) Theorem 1.10 (Bezout’s´ identity). Let a and b be integers, not both 0. Let d = (a, b). There exists integers u, v 2 Z such that d = au + bv. Remark 1.11. This is another application of well ordering principle! Proof. Let S = fam + bn : m, n 2 Z, am + bn > 0g 1) Show that S is nonempty. 2 2 2 2 Take m = a, n = b, then a + b = aa + bb > 0. Since a, b are not both zero, a + b > 0. Now apply the Well-Ordering Principle to get t = au + bv 2) We now wish to show t = (a, b). We must first show that tja and tjb. By the Division Algorithm, a = tq + r, 0 6 r < t Now we can rewrite this as r = a - tq = a -(au + bv)q = a - auq - bvq = a(1 - qu) + b(-vq) Thus r 2 S and r < t. By our choice of t this forces r = 0. A similar argument shows that tjb. 3) We must show that t is greatest divisor. Let cja and cjb. Thus a = kc and b = k0c. Then t = au + bv = kcu + k0cv = c(ku + k0v). Then jtj = jku + k0vjjcj Thus c 6 jcj 6 jtj = t (where the last follows as t > 0) 1.3. Primes and Factorization. ...[prime numbers] are fundamental. They’re the atoms of arithmetic. Just as the Greek origin of the word ‘atom’ suggests the primes are ‘a-tomic’, meaning ‘uncuttable, indivisible.’ And just as everything is composed of atoms, every number is composed of primes. – Steven Strogatz5 Primes are the building blocks of Z. Primes determine structure (e.g. Fundamental Theorem of Arithmetic). Primes are ‘atomic’ or ‘irreducible’. Definition 1.12. An integer p is prime if p 6= 0, ±1 and the only divisors of P are ±1 and ±p.