Lecture notes on Abstract Algebra Uli Walther c 2021 Version of Spring 2021 Contents Basic notions 7 0.1. How to use these notes 7 0.2. Set lingo 7 0.3. Size of sets 8 0.4. Finite vs infinite 9 0.5. Inclusion/Exclusion 10 Chapter I. Week 1: Introduction 13 1. Induction 13 1.1. Setup 13 1.2. The idea 13 1.3. Archimedean property and well-order 16 2. Arithmetic 18 2.1. The Euclidean algorithm 19 2.2. Primes and irreducibles 21 2.3. Some famous theorems and open problems on prime numbers 23 3. Modular arithmetic 26 3.1. Computing \modulo": Z=nZ 26 3.2. Divisibility tests 27 Chapter II. Week 2: Groups 29 1. Symmetries 29 2. Groups 30 3. Cyclic and Abelian groups 32 4. Automorphisms 34 5. Free groups 36 Chapter III. Week 3: Z=nZ and cyclic groups 37 1. Subgroups of cyclic groups 37 2. Products and simultaneous modular equations 39 3. U(n): Automorphisms of Z=nZ 41 Chapter IV. Week 4: Cosets and morphisms 45 1. Equivalence relations 45 2. Morphisms 45 3. Cosets for subgroups 47 4. Kernels and normal subgroups 49 Chapter V. Week 5: Permutations and the symmetric group 51 Chapter VI. Week 6: Quotients and the Isomorphism Theorem 57 3 4 CONTENTS 1. Making quotients 57 2. The isomorphism theorem 60 Chapter VII. Week 7: Finitely generated Abelian groups 63 1. Row reduced echelon form over the integers 63 2. Generating groups 66 Chapter VIII. Week 8: Group actions 71 Review 77 Chapter IX. Week 9: Introduction to rings 79 Chapter X. Week 10: Ideals and morphisms 83 Chapter XI. Week 11, Euclidean rings 87 1. Euclidean rings 87 Chapter XII. Divisibility, Field Extensions 93 1. Divisibility 93 2. Making new fields form old 96 Chapter XIII. Splitting fields and extension towers 99 1. Roots with multiplicity 100 Chapter XIV. Week 14: Minimal polynomials and finite fields 103 1. Minimal Polynomials 103 2. Finite Fields 105 Chapter XV. Galois 109 1. The Frobenius 109 2. Symmetries 112 3. Applications 112 Stuff for later 113 3.1. Zerodivisors 113 3.2. Cartesian Products, Euler's φ-function, Chinese Remainder 115 3.3. Fermat's little theorem 118 CONTENTS 5 Expected progress: • Week 1: Archimedes, Factorization • Week 2: Symmetries, Groups, Subgroups, Order, Aut(G) • Week 3: Z=nZ, Products, U(n) • Week 5: Cosets, Morphisms • Week 4: Symmetric and Free Group • Week 6: Normal Subgroups, Quotients, Automorphism Theorem • Week 7: Finitely Generated Abelian Groups • Week 8: Review, Group Actions • Week 9: Reading day, Intro to Rings • Week 10: Midterm, Ideals, Morphisms • Week 11: Euclidean Algorithm, PID, UFD • Week 12: Fields, Eisenstein, Extensions, • Week 13: Degrees and Splitting Fields • Week 14: Minimal Polynomials, Finite Fields • Week 15: Galois Outlook, Review Basic notions 0.1. How to use these notes. These notes contain all I say in class, plus on occasion a lot more. If there are exercises in this text, you may do them but there is no credit, and you need not turn them in. All exercises that are due are specifically listed on gradescope. This initial chapter is here so we have acommon understanding of the basic symbols and words. This should be known from MA375 (at least if I teach it). Chapter 1 is still more than we did in week 1, but almost all of it should be familiar, and the rest (the open problems on primes) is for your entertainment. Future chapters correspond to actual weeks of classes and are much less verbose than the Basic Notions. The chapter “Stuff for later" can be ignored. It will be worked in when the time comes. Remark .1. There are typos in these notes. If you find some, please inform me. 0.2. Set lingo. The mathematical word set denotes what in colloquial life would be called a collection of things. The \things" are in mathematics referred to as elements of the set. If S is a set and s is an element of S then one writes s 2 S. A sub-collection S0 of elements of S is a subset and one writes S0 ⊆ S, allowing for the possibility of S0 being all of S, or to have no element. There is, strangely, a set that contains nothing. It's called the empty set (denoted ?) and, despite its humble content, one of the most important sets. One uses the notation S = fs1; : : : ; sn;:::g to indicate that S consists of exactly the elements si. (In many cases, one must allow the index set to be different from N. In other words, not all sets can be \numbered", a fact we explore a bit below). A function φ: A ! B from the set A to the set B is an assignment (think: a black box) that turns elements of A into elements of B. The crucial conditions are: the assignment works for every single input a 2 A (so the black box does not choke on any input from A) and for each input there is exactly one output specified (no more, no less). Graphically, functions are often depicted by the help of arrows (starting at the various elements of A and ending at the value φ(a) for each input a). (For an example, suppose φ: Z ! Z is the process of doubling. Then one could write φ(3) = 6, or 3 7! 6). The set A is usually referred to as source, the set B as target. Definition .2. The function φ: A ! B is (1) surjective (\onto") if every element b appears as output of φ; (2) injective (\into") if the equality of outputs φ(a1) = φ(a2) occurs exactly when the inputs a1 and a2 were equal; (3) bijective if it is injective and surjective. 7 8 BASIC NOTIONS An injective map is indicated as A,! B, a surjective one by A B. For example, the function φ(x) = x3 is a bijection from A = R to B = R (all real numbers have exactly one cubic root); the function φ(x) = x2 is neither injective (since always φ(x) = φ(−x)) nor surjective (since negative numbers have no real roots). We will often say \map" instead of \function". 0.3. Size of sets. We wish to attach to each set S a size denoted jSj. In order to make sense of this, we need to compare sets by size. Definition .3. We write jSj ≤ jS0j if there is an injective map φ: S,! S0. Do not confuse the symbols ≤ and ⊆. The following examples illustrate the nature of the relation ≤. Example .4. jNj ≤ jZj since each natural number is an integer. Exercise .5. Show that jZj ≤ jNj. Example .6. • jZj ≤ jQj since each integer is a rational number. • jQj ≤ jRj since each rational number is also real. • Somewhat shockingly, jQj ≤ jZj. To see this, it will be sufficient to prove that there is a way of labeling the rational numbers with integer labels. (One can then make an injective map that sends each rational to its label). How does one label? Imagine sorting the rational positive numbers into a two-way infinite table, as follows: q . p .. 1 2 3 4 ··· 1 1=1 1=2 1=3 1=4 ··· 2 2=1 2=2 2=3 2=4 ··· 3 3=1 3=2 3=3 3=4 ··· 4 4=1 4=2 4=3 4=4 ··· . .. Clearly all positive rationals appear (multiple times) in the table. Now suppose you are moving through the table \on diagonals" where p + q is constant: start at 1=1, the only square on its diagonal (where p + q = 2). Next go on the diagonal p + q = 3, starting on the square with 1=2 and then moving down and to the left. Next walk along the diagonal p + q = 4 starting on 1=3 and moving down and left. It is clear that this process allows you to label each field: 1=1 is number 1, 1=2 is number 2, 2=1 is number 3, and so on. So, the set of all squares is in bijection with the set N. Since all positive rationals are sorted into the various fields, it follows that jQj ≤ jfall squaresgj ≤ jNj. A similar idea can be used on negative numbers, and this shows that jQj ≤ jZj. • In contrast, the statement jRj ≤ jQj is false. The idea is due Cantor, and goes like this. If you believe that you can inject R into Q then you can also inject R into Z because jQj ≤ jZj. Since jZj ≤ jNj, this also implies that you can inject R into N. To inject R into N means to label the real numbers by using only natural (non-repeated) indices. In particular, this can be done to the reals between 0 and 1. BASIC NOTIONS 9 Suppose we have an exhaustive enumeration (0; 1) = fr0; r1; r2;:::g of all real numbers in the unit interval. Let ri;j be the j-th digit in the decimal expansion of ri. So, ri is the real number with expansion 0:ri;1ri;2ri;3 :::. Now write the real numbers into a two-way infinite table: j . i .. 1 2 3 4 ··· 1 r1;1 r1;2 r1;3 r1;4 ··· 2 r2;1 r2;2 r2;3 r2;4 ··· 3 r3;1 r3;2 r3;3 r3;4 ··· 4 r4;1 r4;2 r4;3 r4;4 ··· . .. We construct now the real number ρ whose decimal expansion is determined as follows: the i-th decimal of ρ is the i-th decimal of ri.