# Algebra, Semester 1, 2017

Office hours: Mon 1:00 - 2:15, Wed 1:00 - 2:15, Fri 1:00-1:30 Room 169, Peter Hall Building

### Recommended Reading

Michael Artin: Algebra

David S. Dummit and Richard M. Foote: Abstract Algebra

Serge Lang: Algebra

### Notes

Thanks a lot to our tutor Jon, who has written up explanations of some of the topics treated in class Jon's notes and thanks a lot to Floyd, who has typed up his notes from class Floyd's notes from class

## Classes

 Week Topics Suggested Reading Assignments Week 1 Examples of matrix groups, different ways to think about the circle group. The finite subgroups of the circle group. Semi-direct products and O(2). Reminder of notions such as cosets, quotient groups, normal subgroups, short exact sequences and the isomorphism theorems. This week and the next I am loosely following the chapter in Artin's book called "finite subgroups of the rotation group". For geometric meaning of determinant, see Determinants Week 2 This week, we continued to work through the material of the same chapter in Artin's book. Some of the concepts from Group theory and linear algebra that we revisited were the orbit-stabilizer formula and the normal form of an element of SO(3). We briefly touched on the quaternions and the three sphere. The three sphere is the group Spin(3) and forms a central extension of SO(3) by the two element group. We will learn more about central extensions and the quaternions in the coming weeks. Review your favorite texts on Lagrange's theorem and different ways to think about group actions. Review the notion of kernel and how it relates to injectivity. If you want to get more familiar with examples of finite groups, you might wish to look at Artin's chapter on the groups of order twelve. If you have not already done so, practice to calculate with permutations. Week 3 Rings, modules, ideals, fields and vector spaces. The cyclic group of order four as a central extension of the cyclic group of order 2 by itself. Abstract polynomials versus polynomial functions. For the definition of central extension, read Jon's notes or Chapter 1.2 of Patrick Nicolls Thesis. Solutions for Tutorial 1 The Wikipedia pages on the Binary Tetrahedral Group, the Binary Octahedral Group, the Binary Icosahedral Group, and the Unit Quaternions and why the latter form the group \$Spin(3)\$. Week 4 Examples of rings: The integers, their quotients, the Gauss integers, polynomial rings, endomorphisms rings, matrix algebras. The rational, the real, and the complex numbers. Two definitions of modules. Ideals. Algebras. Ring homomorphisms and R-linear maps. Vector spaces. Quotient rings. A subset of a ring is a two-sided ideal if and only if is the kernel of a ring-homomorphism. Generating sets. The ideal generated by a subset S of R (first definition as the intersection of all ideals containing S). How to define a group homomorphisms, when the source is given by generators and relations. Ben was right: Let X be a set of generators of G and let W be the set of words encoding the relations between these generators. In order to express G quotient of the free group Fr(X), we need to form the normal closure of W inside Fr(X). This is the smallest (by inclusion) normal subgroup of Fr(X) containing W. (By the same token as in Wednesday's class, the normal closure is the intersection of all normal subgroups containing each of these words). To get a good grasp of the definitions of the ideal generated by a subset X of a ring R (or, more generally, the submodule generated by a set inside a module M), you might wish to review the concept of Span of a subset X inside a vector space V from Linear Algebra. Solutions for Tutorial 2 Assignment 1. Thanks to the student who shared his Solutions to Assignment 1. Here are Jon's Solutions and Comments including an email by Arun about writing. Week 5 Free objects: free groups, free abelian groups, free modules, group algebras, free R-algebra, free commutative R-algebra. Generators and relations revisited. Arun's "go-to" recipe for defining group homomorphisms and isomorphisms if your groups are given to you in terms of generators and relations. Universal property of free objects. Solutions for Tutorial 3 Here is the Wikipedia page Presentation of a group Week 6 More examples, twisted group algebras. The complex numbers, the quaternions, and the 3-sphere revisisted. Intergral domains and Principal ideal domains. The structure theorem for finitely generated abelian groups. Noetherian rings. The Smith normal form. Operations on ideals (to be revisited). We followed Wikipedia for the Smith normal form. Here are some useful examples from Jon's Tutorial and the story of the twisted group algebra from my tutorial. Week 7 On Monday we proved that the three conditions characterizing Noetherian rings are indeed equivalent: (1) ascending chain condition on ideals (2) every ideal finitely generated (3) every submodule of a finitely generated module is again finitely generated. On Wednesday we did a mini-review of all the different bits and pieces of theory that went into proving the structure theorem for finitely generated abelian groups. Then we proved the Chinese Remainder Theorem. Week 8 We generalize the proof of the structure theorem, including its reformulation in terms of prime powers (using the Chinese Remainder Theorem) to finitely generated modules over a PID. For this, we need to figure out how to think about concepts like, divisibility, irreducibility, invertibility, greatest common divisor and Bezout's identity in terms of ideals. This is a simple yet powerful application of our two ways of thinking of the ideal generated by two elements a and b: the "funky" definition giving the gcd and the "pedestrian" giving the other side of Bezout's identity. It is a nice little exercise, if you want to prepare for this week. Here is the moment to appreciate the meaning of operations on ideals, and we will briefly touch on this. Then we focus on the example of the polynomial ring in one variable over the complex numbers. Two very big theorems follow as direct corollaries of the structure theorem for modules: the Jordan Normal Form Theorem and the Cayley-Hamilton Theorem over the complex numbers. Week 9 Field extensions, characteristic, finite fields have prime power order, prime ideals. Tutorial questions this week are about the implication PID => UFD. The goal is to review and practice the defintions of the last few classes, but you are also filling in the gaps I left in the proof of the structure theorem over an arbitrary PID. 1. Spell out under which condition on p the principal ideal generated by p is prime. Such a p, if it is non zero, is called a prime in R. 2. Let p be a prime in R. Show that p is irreducible. 3. In a PID, the converse also holds: irreducible elements are prime. Why? 4. Show that in a PID, all non zero prime ideals are maximal. 5. Find a non maximal prime ideal in your favourite non PID. 6. Let R be a PID. Show that every element a of The possesses an irreducible factor. 7. Then show that a can be written as a finite product of irreducible factors. 8. Formulate and prove uniqueness of this factorisation. 9. Now prove that in a PID we have lcm(a,b)=ab/gcd(a,b) and deduce the Chinese remainder theorem. Week 10 Constructible numbers Tutorial questions for this week (and next): Let F be a field, K a field extension of F and L a field extension of K. (1) Show that the degree of field extensions is multiplicative in the following sense: [L:F] = [L:K] [K:F]. (2) We call the extension K of F algebraic if each element of K is algebraic over F. Show that every field extension of finite degree is algebraic. (3) Assume that K is algebraic over F and that a is an element of L which is algebraic over K. Deduce that a is also algebraic over F. (4) Let R be an integral domain. Familiarise yourself with the definition of field of fractions of R by working through the relevant defintions and proofs. Why did I demand that R be an ID? Assignment 2 Thanks for DJ for sharing his Solutions for Assignment 2. Week 11 Finite fields tutorial questions part 1 tutorial questions part 2