NUMBER THEORY VIA ALGEBRA AND GEOMETRY DANIEL LARSSON CONTENTS 1. Introduction 2 2. Rings 3 2.1. Definition and examples 3 3. Basics @ring 5 3.1. Ideals and subrings 6 4. Integral domains 7 4.1. Homomorphisms, quotient rings and the first isomorphism theorem 8 4.2. UFD’s, PID’s and Euclidean domains 13 4.3. The Gaussian integers 18 4.4. Polynomials 20 5. Fields 22 5.1. Definition and examples 22 6. Basics @field 23 6.1. Fields of fractions 24 7. Field extensions 25 7.1. Field extensions 25 7.2. Algebraic extensions, transcendental extensions 26 7.3. Simple and finitely generated extensions 27 7.4. Algebraic closure 28 8. Finite fields 28 8.1. The main theorem 29 8.2. The Frobenius morphism 30 9. Algebraic number fields 30 9.1. Algebraic numbers 30 9.2. Norms, traces and conjugates 31 9.3. Algebraic integers and rings of integers 34 9.4. Integral bases 38 9.5. Computing rings of integers 39 9.6. Examples 42 10. Quadratic number fields 44 10.1. Ring of integers of quadratic number fields 44 10.2. The Ramanujan–Nagell theorem 44 11. Dirichlet’s unit theorem 47 11.1. Roots of unity 47 11.2. Units in number fields 48 12. Dedekind domains 49 12.1. A few important remarks 49 12.2. The main theorem on Dedekind domains 50 1 2 D. LARSSON 13. Extensions, decomposition and ramification 54 13.1. Ramification and decomposition 55 13.2. Consequences for quadratic number fields 58 13.3. Consequences for some non-quadratic number fields 64 14. Cyclotomic number fields 65 14.1. Cyclotomic fields 65 14.2. Galois theory of number fields 72 14.3. Gauss sums and Quadratic Reciprocity 74 14.4. Cubic reciprocity 81 15. Arithmetic and Geometry 84 15.1. Affine n-space 84 15.2. Projective n-space 86 15.3. Algebraic curves 87 15.4. Cubic and elliptic curves 91 15.5. The group structure of an elliptic curve 92 15.6. The group law 93 15.7. Points of finite order 95 15.8. The Nagell–Lutz theorem 96 15.9. Mordell’s Theorem and Conjecture 97 16. Gauss’ Class Number Problem and the Riemann hypotheses: A Historical Survey 98 16.1. Gauss’ class number problem 98 16.2. Quadratic fields and forms 99 16.3. What does this have to do with the class number problem? 103 16.4. Zeta functions and the Riemann hypothesis 104 16.5. Back to quadratic fields 107 17. Appendix A: Linear algebra 109 17.1. Vector spaces and bases 109 17.2. Maps 109 17.3. Dual spaces 110 17.4. Operations on maps 110 17.5. Linear equations and inverses 111 17.6. Modules 111 17.7. Vandermonde determinants 112 18. Appendix B: Chinese remainder theorem 113 1. INTRODUCTION These set of notes are the Lecture Notes accompanying my class in Number theory at Uppsala University, Spring terms 2008 and 2009. The notes begin with some very basic ring and field theory in order to set the stage, and continues to more advanced topics successively and probably in a rather steep upwards slant from an extremely soft and cosy start. I allow myself a few digressions in the text that are not part of the syllabus but that I feel are in a sense part of the required “know-of” for aspiring mathematicians. In this case I am mainly refering to the sections concerning the Gauss’ class number problem and the Ramanujan–Nagell theorem. These are then not formally part of the course and will not 3 be discussed in an exam. But I strongly encourage readers to at least read through these parts to get an idea of the beauty that lies within. Also, some parts are more abstract and technical, mainly the section on rings of in- tegers. The proofs here are rather difficult but I felt that if I didn’t include them in the course I would be cheating (which I don’t like). Therefore, I don’t require the students to learn this material, but rather to have this as a fall-back solution if the later results feel a bit hollow and improperly motivated. In the same spirit, I include Appendices with notions from “linear algebra over rings” for easy reference. As a twist to this course I added a section on elliptic curves, a topic that, without a doubt, will be part of every course on number theory that ever will be given anywhere on the planet, or elsewhere (this is a foretelling on my part). A home-assignment will be given where the students are to learn the basics of elliptic curves over finite fields, so that they immediately can understand the basic ideas and quickly learn the techniques of elliptic curve cryptography and (large) integer factorization using elliptic curves. This will surely be a worthwhile effort for every mathematically inclined student. 2. RINGS I will assume that everyone knows what an abelian group is. 2.1. Definition and examples. We begin with the following definition. Definition 2.1. Let R be a binary set with two closed operations, ’+’ (addition) och ’∗’ (multiplication). Then R = (R;+;∗) is a ring if addition and multiplication is compatible according to the following axioms: Rng1: The operation + makes R into an abelian group, that is, a + b = b + a for all a;b 2 R. Rng2: There is an element 1 such that 1 ∗ a = a ∗ 1 = a for all a 2 R, or in other words, a multiplicative unit, often simply called a one. Rng3: The multiplication is associative: a ∗ (b ∗ c) = (a ∗ b) ∗ c; 8a;b;c 2 R: Rng4: The multiplication is distributive: a ∗ (b + c) = a ∗ b + a ∗ c and (b + c) ∗ a = b ∗ a + c ∗ a; 8a;b;c 2 R: Remark 2.1. Strictly speaking, what we have defined here should be called an associative ring with unity (or associative, unital ring) The most general definition includes only Rng1 och Rng4. There are many examples of structures satisfying only these two axioms. However, for us it is enough to state the definition in the above, more restrictive, way. Note! From now on we write multiplication in the usual fashion ’a · b’ or simply ’ab’. 2.1.1. Examples. Example 2.1. The easiest and most obvious (and arguably the most important) examples are of course the following - Z, the ring of integers; - Q, the ring of rational numbers; - R, the ring of real numbers; - C, the ring of complex numbers. 4 D. LARSSON Convince yourselves that these are indeed rings! In fact, they are even commutative. Example 2.2. Another, extremely important example of a ring is Z=hni, the ring of integers modulo n. Recall that this is the set of congruence classes of elements modulo division by n, and can be represented by f0;1;2;:::;n − 1g, the remainders after division with n. Informally, one writes Z=hni := f0;1;2;:::;n − 1g; implicit being that addition and multiplication are allowed. Example 2.3. Another example is p p Z[ n] := fa + b n j a;b 2 Zg for n a square-free integer. We will see a lot of examples like this in the future, since number-theoretic applications to rings often involve examples like this one. So far, every example has been commutative. Let’s round off this example-listing with a non-commutative number system. Example 2.4 (Quaternions)p. Recall that the complex numbers C was formed by, to R, add an imaginary number i := −1. As you no doubt remember C := fa + bi j a;b 2 Rg: The ring C has one further property that we haven’t discussed yet (but will, in the next 1 section): every non-zero complex number has an inverse A question that was askedp during around if one could add yet another “really” imaginary element (not equal to −1), j, so that one gets a new ring with the property that every non-zero element has an inverse. This can be shown to be impossible! However, 1843, W.R. Hamilton realized that if one adds two more elements to C, the element j and another k, then every non-zero element has an inverse! But, alas, one loses one important thing: commutativity. The definition is as follows. Definition 2.2. The set of all “numbers” 2 2 H := fz = a + bi + cj + dk j a;b;c;d 2 R; i = j = −1 · 1; ij = −ji = kg is a non-commutative ring, where all elements 6= 0 has an inverse. An element of H is called a quaternion and H is called the ring of quaternions. I strongly suggest that you do the following exercise. Exercise 2.1. Try out a few computations and check that H is indeed a non-commutative ring. (You don’t have to check that it is a ring. This is obvious! Why?) To define an inverse recall how inverses are computed in C and emulate that. In order to do this you will have to define a suitable notion of conjugate quaternion. Notice that R ⊂ C ⊂ H and that we double the dimension in each step: C is two- dimensional as a vector space over R and H, is four-dimensional. The natural follow-up question is: is it possible to add more imaginary elements and get larger number systems? The answer is yes, and this was given only a few months after Hamilton’s discovery by J.T Graves and later by A.