# Algebra

## Class

### Recommended Reading

Michael Artin: Algebra

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

Serge Lang: Algebra

Thanks to Cohen, we now have Lecture Notes. You will need to log in from your unimelb account.

The construction and uniqueness (up to non-unique isomorphismthis time) of the field with p^n elements.
 Week Topics Resources Tutorials and Assignments 1 Review of advanced topics in group theory at the example of the tetrahedral group (alternating group on four elements): normal subgroups and quotient groups, universal property of the quotient group, conjugate subgroups, Sylow theorems, stabilizers. The 2017 version of this class. Thanks to Angus for sharing these links to blog posts about quotient groups, more on quotient groups and the first isomorphism theorem. Tutorial 1 and Solutions 2 Rings, algebras, ideals, fields, characteristic, and field extensions. Endomorphism algebras. Integral domains and their fields of fractions. Notes for Arik Wilbert's Lecture (Lecture Capture for this lecture was audio only, sorry) Tutorial 2 3 Modules and vector spaces. Any finite field has prime power order. Group algebras. Representations (in the assignment). The example of the tetrahedral group. The group algebra construction as a free functor (universal properties revisited, in the assignment). Polynomial rings and their quotients. Tutorial 3 and Solutions and Assignment 1 due on Friday of Week 6 in class. Many thanks to the students who have shared their solutions: Sample solutions 1 Sample solutions 2 4 Polynomial rings, ideals and field extensions: We are headed towards constructing extension fields. The recipy for this is to start with a polynomial ring and take the quotient by the principal ideal generated by some polynomial. The question is now "under which circumstances is this quotient ring a field?" And the answer will be "if and only if the polynomial in question was irreducible". Examples of this scenario include our constructions of GF(8) from GF(2), of GF(4) from GF(2), and of the complex numbers from the real numbers. It is fairly easy to see that irreducibility is a necessary condition (otherwise you would get zero divisors in the quotient). Currently we are working our way towards understanding under which conditions irreducibility implies that the quotient is a field. For this we need to establish some vocabulary: We will need maximal ideals, prime ideals, irreducible elements and how they are all related, and Euclidean domains, principal ideal domains, unique factorization domains, gcd domains. For those who want to learn more about universal properties, MacLane's book "Categories for the working mathematician" is an exellent resource. Here is an old activity sheet I made for the opening week of my category theory class last year. Others have asked about representation theory. Excellent introductory reading is Serre's book "Linear representations of finite groups" or the book by James and Liebeck with a similar title. Tutorial 4 5 Domain hierarchy: Euclidean domains, PIDs and UFDs. Maximal ideals and prime ideals, irreducible elements. Constructions with straight edge and compass. Tutorial 5 Solutions 6 Constructions with straight edge and compass continued. Algebraic and transcendental elements of a field extension. The degree of a field extension. Tutorial 6 Solutions Assignment 2 released, due in Week 9 Sample solutions 1 7 The remaining proofs about the domain hierarchy. Putting everything together, we have now seen that in a Euclidean domain, any irreducible element generates a maximal ideal. In particular, Z/pZ is a field if p is a prime number (this you already knew) and k[x]/(p(x)) is a field if k is a field and p is an irreducible polynomial over k. Revision and some preview. Tutorial 7 Solutions 8 Splitting fields Tutorial 8 9 The Galois correspondence Monday Lecture Tutorial 9 and selected solutions and Friday Lecture 10 Modules over principal ideal domains. We have already seen that modules over the integers are the same as abelian groups, modules over the polynomial algebra k[x] for k a field will be important for the Jordan Normal Form Tutorial 10 and Assignment 3, due on the last day of classes 11 The Smith normal form Lecture by Alex 12 As applications of the Smith normal form, we obtain the Jordan normal form and the structure theorem for finitely generated abelian groups. End semester review.

## Undergraduate Seminar

This year, we will be offering an companion seminar on Coxeter groups and Hecke algebras, organized by Arik Wilbert and Gufang Zhao. This is aimed at the level of the Algebra students, but everybody is welcome to participate. Talks will be for 100 minutes with a break in the middle. Participants will prepare their presentations under the guidance of Arik and Gufang and give a practice talk in advance. Great emphasis is put on clear and compelling exposition.

The seminar will be meeting weekly throughout the semester on Thursday afternoons from 3:15 to 5:15 in Russel Love Theatre, Peter Hall Building.

 Week Date Topics Speaker 1 7 March Algebraic preliminaries: Free groups and free algebras, generators and relations Arik Wilbert 2 14 March The Symmetric group and its Coxeter presentation: [KT] Sections 4.1.1-4.1.3. William Mead 3 21 March Notes by Jonah Reduced expressions and exchange theorem: [KT] Section 4.1.4 (state and prove Lemma 4.6 and 4.7 and use them to prove Theorem 4.8 which is the main result for this week; finish with Corollary 4.9 and Lemma 4.10 which we will need in week 5). Jonah Nelson 4 28 March Equivalences of reduced expressions: [KT] Section 4.1.5 (state and prove Lemma 4.11 and Theorem 4.12). Cara Faulkner 5 4 April The standard basis of the Iwahori-Hecke algebra: [KT] Sections 4.2.1 and 4.2.3, skip 4.2.2 or (if you are ambitious) rewrite Sections 4.2.1 and 4.2.3 in terms of the single-variable Hecke algebra in the first place (Definition 4.15, Lemma 4.16, Theorem 4.17, Lemmas 4.18, 4.19 and 4.20) Matthew Bunney 6 11 April The Kazhdan-Lusztig basis of the Iwahori-Hecke algebra: [S] Read the introduction and Section 2 and give a detailed proof of Theorem 2.1; compute the Kazhdan-Lusztig polynomials for the Hecke algebra of the symmetric group on three letters. Tom Cummings 7 18 April Survey on traces, HOMPLY-PT polynomials and link invariants Finn McGlade 8 2 May Finite groups of isometries (rigid motions), dihedral groups. Reference [A] Chapter 4, Section 5 and Chapter 5, Sections 2-3. Time permitting: the circle group and the matrix group O(2). Will McDonald and Liam Caroll 9 9 May Coxeter groups and examples Reference on the definition of Coxeter groups: [H] Section 1.1. Reference for finite groups of isometries of R^3 [A] Chapter 5, Section 9 (study the proof but only present the statement of the theorem). Jackson 10 16 May The matrix groups SU(2) and SO(3) and their finite subgroups. Reference: [A] Chapter 8, Section 2-3. Rohan Hitchcock and Brett Eskrigge 11 23 May An expository lecture on configuration spaces. Adrian Putra References [A] M. Artin: Algebra, first edition, Englewood Cliffs, N.J. : Prentice Hall, c1991. xviii, 618 p. Note: if you use other editions, the section numbers might be different than listed below. [KT] C.Kassel and V.Turaev: Braid Groups [H] J.Humphreys: Reflection groups and Coxeter groups [S] W.~Soergel: Kazhdan-Lusztig polynomials and a combinatoric for tilting modules, Rep Theory 1 (1997), 83-114