Introduction to Abstract Algebra \Rings First" Bruno Benedetti University of Miami January 2020 Abstract The main purpose of these notes is to understand what Z; Q; R; C are, as well as their polynomial rings. Contents 0 Preliminaries 4 0.1 Injective and Surjective Functions..........................4 0.2 Natural numbers, induction, and Euclid's theorem.................6 0.3 The Euclidean Algorithm and Diophantine Equations............... 12 0.4 From Z and Q to R: The need for geometry..................... 18 0.5 Modular Arithmetics and Divisibility Criteria.................... 23 0.6 *Fermat's little theorem and decimal representation................ 28 0.7 Exercises........................................ 31 1 C-Rings, Fields and Domains 33 1.1 Invertible elements and Fields............................. 34 1.2 Zerodivisors and Domains............................... 36 1.3 Nilpotent elements and reduced C-rings....................... 39 1.4 *Gaussian Integers................................... 39 1.5 Exercises........................................ 41 2 Polynomials 43 2.1 Degree of a polynomial................................. 44 2.2 Euclidean division................................... 46 2.3 Complex conjugation.................................. 50 2.4 Symmetric Polynomials................................ 52 2.5 Exercises........................................ 56 3 Subrings, Homomorphisms, Ideals 57 3.1 Subrings......................................... 57 3.2 Homomorphisms.................................... 58 3.3 Ideals.......................................... 60 3.4 Generators and Principal Ideals............................ 62 3.5 PIDs........................................... 64 1 3.6 Noetherian domains................................... 66 3.7 Exercises........................................ 70 4 Quotients 71 4.1 Equivalence relations.................................. 71 4.2 Quotient Ring...................................... 72 4.3 Homomorphism theorems................................ 74 4.4 *Fields of fractions................................... 78 4.5 Exercises........................................ 80 5 Prime and maximal ideals 81 5.1 Operations with ideals................................. 81 5.2 Prime ideals and quotients............................... 83 5.3 Maximal ideals and quotients............................. 84 5.4 Radical Ideals and Nilpotent Elements........................ 86 5.5 Exercises........................................ 89 6 Factorizations 90 6.1 Irreducible elements.................................. 90 6.2 UFDs.......................................... 96 6.3 *Gauss' theorem.................................... 99 6.4 Irreducibility in Z[x].................................. 104 6.5 Exercises........................................ 112 7 Field extensions 113 7.1 Algebraic and transcendental numbers........................ 114 7.2 Field extensions and degree.............................. 117 7.3 Algebraic closure and Nullstellensatz......................... 122 7.4 The fundamental theorem of algebra......................... 126 p 7.5 *Q( p) and Fibonacci numbers............................ 130 7.6 Exercises........................................ 134 8 Groups 136 8.1 Groups, subgroups, homomorphisms......................... 136 8.2 Lagrange theorem, normal subgroups, quotients................... 139 8.3 Generators and period of an element......................... 143 8.4 Cyclic and finitely generated groups......................... 145 8.5 Abelian Groups, Chinese remainders, and the totient function........... 148 8.6 Permutations...................................... 155 8.7 *Simple and Solvable Groups............................. 159 8.8 *Fundamental theorem of finitely generated Abelian groups............ 164 8.9 Exercises........................................ 172 9 Galois Theory 173 9.1 Degree-three equations................................. 174 9.2 Degree-four equations................................. 176 9.3 Degree-n equations with only two nonzero terms.................. 177 9.4 The Galois group.................................... 179 9.5 The Galois Correspondence.............................. 186 9.6 Equations that cannot be solved by radicals..................... 190 2 9.7 Exercises........................................ 194 10 Modules 195 10.1 Modules, submodules, quotients and direct sums.................. 195 10.2 Homomorphisms, short exact sequences, and projective modules......... 196 10.3 Homology and free resolutions............................. 200 10.4 *Smith normal forms and finitely generated modules................ 202 3 0 Preliminaries In this section we briefly recall a few topics you probably know already. But even if you \know too much", it's good to agree on notation and terminology, so that we are all on the same page. 0.1 Injective and Surjective Functions Definition 1. A function f : X −! Y consists of two sets X; Y and a subset F ⊂ X × Y , such that for each element x 2 X there is always exactly one element y of Y for which (x; y) 2 F . Usually we denote this y by f(x), and we say it is the image of x (under f). We also call X (resp. Y ) the domain (resp. the codomain) of the function. The image of the set X is the set Im X =def ff(x) such that x 2 Xg. Functions are typically described by a formula that tells us how to find f(x) given x. For example, given any set X, the identity function on X (usually denoted by idX ) is the function whose output is always identical to the input. In this case, our notation to \explain the function" is idX : X −! X x 7−! x: Sometimes one does not have an explicit formula, but there is still a clear general method to associate x with its image: for example, f : N −! N x 7−! the x-th prime number: In the worst case scenario, if we do not see a general pattern, we can always express f by specifying all its values: f : f0; 1; 2g −! f0; 1; 2g 0 7−! 1 1 7−! 0 2 7−! 2: Definition 2. A function f : X ! Y is injective if for each x 6= x0 one has f(x) 6= f(x0). We assume familiarity with logic, in particular with quantifiers (8, 9) and logical equivalence (contrapositives, etc.) For example, it should be clear that an equivalent way to define injectivity is: 8x; x0 2 X; f(x) = f(x0) ) x = x0: Is it clear? Or does it look like hieroglyphics? If you are lost, please revise your notes from the prerequisite courses before going on... Injectivity depends not only on the \formula", but also on the domain involved. For example, f : −! f : −! N N is injective, Z N is not. x 7−! x2 x 7−! x2 Definition 3. A function f : X ! Y is surjective if the image of X coincides with the codomain Y ; or in other words, if for each y 2 Y there is some x 2 X (not necessarily unique) such that y = f(x). 4 Surjectivity depends not only on the \formula" for f, but also on the domain and the codomain. For example, let R>0 (resp. R≥0) be the positive (resp. the nonnegative) real numbers: f : −! f : −! f : −! R R is not surjective, R R≥0 is, R>0 R≥0 is not. x 7−! x2 x 7−! x2 x 7−! x2 Definition 4. A function f : X ! Y is bijective if it is both injective and surjective. That is, if for each y 2 Y there exists exactly one x 2 X such that y = f(x). Given a function f : X ! Y and a function g : Y ! Z, their composite is the function g ◦ f : X −! Z x 7−! g( f(x)): Proposition 5. Let f : X ! Y be a function between two non-empty sets. (1) f is surjective () there exists g : Y ! X (called \right inverse") such that f ◦ g = idY . (2) f is injective () there exists g : Y ! X (called \left inverse") such that g ◦ f = idX . (3) f is bijective () there exists g : Y ! X (called \inverse") such that g ◦ f = idX and f ◦ g = idY . Remark 6. Before starting with the proof, note that two functions are equal when they have same domain, same codomain, and they yield same outputs when given same inputs. So to verify an equality of functions like g ◦ f = idY , both going from Y to Y , we'll need to check that g ◦ f(y) = idY (y) for all y 2 Y . Proof of Proposition5. (1), \)". Define g : Y −! X y 7−! some x such that f(x) = y: (If there is more than one x such that f(x) = y, we simply choose one.) Then by con- struction, f ◦ g(y) = f(x) = y for all y 2 Y . Hence f ◦ g = idY . (1), \(". For each y 2 Y , we know that idY (y) = f ◦ g(y), so y = f(g(y)), which means y 2 Im f. (2), \)". Choose a point x0 of X. Define g : Y −! X x ; if y2 = Im f y 7−! 0 the unique x such that f(x) = y; if y 2 Im f: Then for all x in X, g ◦ f(x) = g(f(x)) = x. So g ◦ f = idX . 0 (2), \(". Suppose f(x) = f(x ). Applying g, and remembering that g ◦ f = idX , we get 0 0 0 0 x = idX (x) = g ◦ f(x) = g(f(x)) = g(f(x )) = g ◦ f(x ) = idX (x ) = x : (3), \)". This does not follow immediately from items (1) and (2), because a priori it could be that the two g's (right inverse and left inverse) are different. However, if f is bijective we can simply define g : Y −! X y 7−! the unique x such that f(x) = y: and it is easy to see that it does the trick. (3), \(". This follows from (1) and (2). (Why?) 5 0.2 Natural numbers, induction, and Euclid's theorem You are probably all familiar with the infinite set of natural numbers (also known as \nonnegative integers") N = f0; 1; 2; 3; : : : ; n; n + 1;:::g It is usually stated in textbooks that natural numbers \come from Nature". This is not entirely true: To accept them, three important abstraction steps are necessary. These steps are non- trivial, as throughout the history of mankind, not all populations have achieved or accepted them: • the notion of cardinality, i.e. the realization that two finite sets in bijection with one another have something in common; whence the names of numbers. This is not universal: Even today, the Pirah~apeople in Amazonas, Brazil, have no names for numbers, and have no linguistic way of expressing exact quantity, not even \one". 1 • the notion of zero, as the cardinality of an \empty set". The ancient Greeks had no symbol for zero, for example; Mayas did have a symbol for zero around the year 36 BC, using it as placeholder in their base-20 numerical system; arithmetic operations with zero were first introduced by the Indian mathematician Brahmagupta2, around 650 AD.