Arun Ram: Algebra Sem I 2024

MAST30005
Algebra

Semester I 2024

Lecturer: Arun Ram, 174 Peter Hall Building, email: aram@unimelb.edu.au

Time and Location:

Lecture: Monday 12:00-13:00, Tuesday 11:00-12:00, Thursday 9:00-10:00 in Russell Love Theatre, Peter Hall Building
Practicals: Monday 15:15-16:15, Monday 16:15-17:15, Wednesday 16:15-17:15 in Room 213, Peter Hall Building

Some of my thoughts about teaching have been written down in the Lecture script: Teaching Math in the Next Life
I am very interested in listening to and discussing any thoughts/reaction/improvements that you have in relation to this content.

The student representatives are
David Lumsden email: dnlumsden@student.unimelb.edu.au
Dhruv Gupta email: dggup@student.unimelb.edu.au

The handbook entry for this course is at https://handbook.unimelb.edu.au/2024/subjects/mast30005/print

Main Topics

(1) Fields, Groups and the Galois correspondence
(2) Integral domains, Euclidean domains, PIDs, ideals and factorisation
(3) Finitley generated 𝔽-modules, ℤ-modules and 𝔽[x]-modules
(4) Generators and relations
(5) Fraction fields and polynomial rings

Announcements

No books, notes, calculators, tablets, ipads, phones, etc at the exam.
Prof. Ram reads email but generally does not respond by email. Usually these are collated and reponses to email queries are provided in the first few minutes of lectures. That way all students can benefit from the answer to the query.
Compared to the other resources on this page Lecture Capture is not an efficient way to absorb material for this course. If you want to get a hang on the naterial covered in a particular lecture, you should make a handwritten copy of the Handwritten Lecture Notes for that lecture which are posted below, and ask any questions that come up as you make your handwritten copy in the Proof and Solution writing sessions.

Proof and Solution writing sessions

Proof and Solution writing sessions are
Tuesdays 12:00-2:00 and Thursday 8:00-9:00 in Russell Love Theatre, Peter Hall Building.

I am often available for additional questions/discussions after class on Monday.

I am available by appointment. I rarely come in to the University before 12 as I have other responsibilities in the morning until 12. If you email me suggesting some weekday afternoon times for an appointment that work for you then I can choose one of them that also works for me. If you email me and don't suggest some times that work for you, then I will respond by asking you to suggest some weekday afternoon times that work for you.

Assessment

Final exam

The problem sheets will be adjusted until the end of the 6th week of class (10 April) after which no further changes will be made to the problem sheets before the exam. The final exam will be constructed by randomly choosing questions from the problem sheets and adjusting for length for a 3 hour final exam.

There will not be special solution sheets for the problem sheets. there are three good ways to check your work:

Go to the Proof and Solution writing sessions and ask for the solution.
Write your solution carefully, give it to someone else and ask them to mark it to help you improve.
Have some friends in the class write the solution carefully and you mark their work to help them improve.

Many solutions are also found in the notes on this web page.

Assignments

Assume that the goal of MAST30005 student is to excel on the exam.
Write a guide for the student to learn the subject and excel on the exam.

Submissions will be marked based on clarity, content, accuracy, thoroughness, quality of mathematical writing, and usefulness for a MAST30005 student for learniing the course material and preparing for the final exam.

Your draft Guide for the Algebra student due: 8 April at 10am.
Submit to Canvas as Assignment 1.
Feedback to each of your peers in your peer marking group due: 15 April at 10am.
For this provide the most helpful written feedback to your peers that you can.
Submit to Canvas as Assignment 2.
Your completed Guide for the Algebra student due: Monday 13 May at 10am. Submission deadline extended to Thursday 16 May at 5pm.
Submit to Canvas as Assignment 3.
Peer marking on Final draft due: 24 May at 10am.
For this make a ranking of the submissions in your peer marking group and write an explanation of the reasons for your ranking.
Submit to Canvas as Assignment 4.

Your final mark on your assignment will be determined by analysis of all four parts of your submission: the draft submission, the final submission, your feedback to your peers on the draft, and your explanation for your ranking on the final submissions for your peer marking group.

Tutorial sheets

There will not be special solution sheets for the tutorials. There are three good ways to check your work:

Go to the Proof and Solution writing sessions and ask for the solution.
Write your solution carefully, give it to someone else and ask them to mark it to help you improve.
Have some friends in the class write the solution carefully and you mark their work to help them improve.

Many solutions are also found in the notes on this web page.

Resources

Recommended are:

The problem sheets and tutorial sheets on this page
The notes on this page
The tutorial sheets and problem sheets on this page
Almost all of the hundreds of books on algebra accessible from the University of Melbourne library and the internet
The problem sheets and tutorial sheets on this page
The notes written by Lawrence Reeves available from The Canvas welcome page for this course.
The notes, tutorial sheets and problem sheets on this page

Another favourite resource for the material we will cover is Chapters 10,11,12,13,14 in the book

Michael Artin, Algebra, Prentice Hall, 1991.

Another favourite resource for the material we will cover is Chapters 7,8,9,10,11,12,13,14 in the book

Dummit and Foote, Abstract Algebra, John Wiley and Sons, 2004.

Notes written by Arun Ram

	Proof machine (How to do proofs)
	Example proofs

Fields	Examples: Fields	Proofs: Fields
Vector spaces	Examples: Vector spaces	Proofs: Vector spaces
Rings	Examples: Rings	Proofs: Rings
Modules	Examples: Modules	Proofs: Modules
Groups	Examples: Groups	Proofs: Groups
Group actions	Examples: Group actions	Proofs: Group Actions

Group families Cyclic groups Dihedral groups Symmetric groups Alternating groups	Commutative rings Fields and integral domains Fields of fractions Polynomial rings Euclidean domains, PIDs and UFDs	Groups of low order $ℤ / 2 ℤ$ , the cyclic group of order 2 $ℤ / 2 ℤ \times ℤ / 2 ℤ$ , the Klein 4 group $S_{3} ≅ D_{3}$ , the nonabelian group of order 6 The dihedral group $D_{4}$ of order 8 The tetrahedral group $A_{4}$ The octahedral group $S_{4}$

Matrix groups Diagonal matrices Permutation matrices Unipotent upper triangular matrices Invertible matrices Matrix group presentations: Some proofs

Galois correspondence	Fields 𝔽(α)	Primitive element theorem
		Primitive element theorem part 2
Finite fields	Cyclotomic polynomials	Splitting fields and algebraic closures
Finite fields Part 2	Cyclotomic polynomials Part 2	Möbius transformations and number fields

Irreducible polynomials	Smith normal form	Finitely generated modules for PIDs
Irreducible polynomials Part 2	Smith normal form Part 2	Finitely generated modules for PIDs part 2

Krull-Schmidt and torsion	PIDs are UFDs	Jordan Normal form

Euclidean domains and PIDs	Maximal and prime ideals	Fractions and Polynomial rings

In class lectures

Part A. Fields

26 February 2024 Lecture 1: Proof machine Hand written Lecture Notes

Student TODO List: Carefully read the notes Proof machine (How to do proofs) and Example proofs before class. After class, do the proof of the proposition presented in class carefully, several times (as if you were practicing a musical instrument). Go through questions 1-48 of the exercises from Problem sheet: Navigation and make a clear note of which ones you need to get more practice/help with before they appear on the exam. Be sure to ask to have any of these that you are the slightest bit unsure about to be done during the Proof and Solution writing sessions this week.
27 February 2024 Lecture 2: The main point of Galois Theory Hand written Lecture Notes

Student TODO List: Carefully read seciton 6.7 of the notes Galois correspondence before class. After class, go through questions 124-146 of the exercises from Problem sheet: Navigation and make a clear note of which ones you need to get more practice/help with before they appear on the exam. Be sure to ask to have any of these that you are the slightest bit unsure about done during the Proof and Solution writing sessions this week. Then do the proof of Theorem 6.16 carefully, several times (as if you were practicing a musical instrument). Be sure to ask to have any parts of this proof that you are the slightest bit unsure about to be done during the Proof and Solution writing sessions this week.
29 February 2024 Lecture 3: Constructing fields between ℚ and ℂ Hand written Lecture Notes

Student TODO List: Carefully read seciton 6.9 of the notes Fields 𝔽(α) before class. After class, go through questions 112-154 of the exercises from Problem sheet: Navigation and make a clear note of which ones you need to get more practice/help with before they appear on the exam. Be sure to ask to have any of these that you are the slightest bit unsure about to be done during the Proof and Solution writing sessions this week. Then do the proof of Theorem 6.19 carefully, several times (as if you were practicing a musical instrument). Be sure to ask to have any parts of this proof that you are the slightest bit unsure about done during the Proof and Solution writing sessions next week. Outline and start filling the details of the section on Fields for your assignment.
4 March 2024 Lecture 4: Theorem of the primitive element Hand written Lecture Notes

Student TODO List: Carefully read section 6.10 of the notes Primitive element theorem before class. After class, read Primitive element theorem part 2. It is, at this point, sensiblle to start becoming familiar with questions 1-97 and 108-199 and 206-252 of the exercises from Problem sheet: Navigation. Make a list of the vocabulary in these questions, and look up or ask in the Proof writing sessions for the definitions of these terms. Make a clear record for yourself of all of these definitions. It might be a good idea to include a vocabulary list containing these definitions in your Assignment.
5 March 2024 Lecture 5: Finite fields and cyclotomic polynomials Hand written Lecture Notes

Student TODO List: Carefully read sections 6.11 and 6.12 of the notes Finite fields and cyclotomic polynomials before class. After class, read Finite fields Part 2 and Cyclotomic polynomials Part 2. Learn how to do the proofs of (each the individual parts of) the theorems from Tutorial 1 NNEW: Last week's theorems quickly, efficiently, and without notes. Practice these regularly so that you can just crunch them out when they appear on the exam.
7 March 2024 Lecture 6: Möbius transformations and algebraic number fields Hand written Lecture Notes

Student TODO List: Read the notes Möbius transformations and number fields. Get a good hack that the writing of the Fields section of your assigment. We have now covered most all of the content for the Fields section of this course. There really is only one theorem: "The Galois correspondence". All the other things in the Fields section of this course are supporting results and examples for this main theorem. We will do further review and examples of the Fields section of this course in the last 6 weeks.

Part B. Modules

11 March 2024 Lecture 7: Irreducibility of polynomials Hand written Lecture Notes

Student TODO List: Read the notes Irreducible polynomials Part 2. Go through the Problem Sheet: Rings and pick out all the questions that ask to determine whether a polynomial is irreducible. Which of these can be done with the Eisenstein criterion and which can't? We have started the section on Modules. The notes Modules will be helpful for solidfying this section. Compare the notes Modules to the notes Vector spaces and find all the analogies between Modules and vector spaces.
12 March 2024 Lecture 8: Reduction to diagonal for PIDS: Smith normal form Hand written Lecture Notes

Student TODO List: Read the notes Smith normal form. Go through the Problem Sheet: Modules and pick out 10 of the questions that ask to find the structure of a module over a PID (i.e. 10 of the questions 3, 18, 26, 27, 30, 55, 56, 57, 58, 71, 72,73, 74, 75, 76, 77, 86, 87, 88, 89, 90, 94, 95, 96, 97, 110, 114, 118, 121, 130, 138, 139, 146, 147, 148, 156, 157, 163, 166, 170, 171, 185, 186, 189, 190, 191, 196, 205, 209) Write out the relevant matrix from each of these questions and reduce it to Smith Normal Form. Use the determinant as a check that you have done the computation correctly. Writing a clear, cogent, accurate exposiion of the Smith Normal Form algorithm can be challenging. Have a go on writing this for your assignment, and discuss how to improve your presentation of this key algorithm with your peers.
14 March 2024 Lecture 9: Finitely generated modules over a PID Hand written Lecture Notes

Student TODO List: Read the notes Finitely generated modules for PIDs. For each of the questions on the Problem Sheet: Modules where you reduced the matrix to Smith Normal form, do the change of generators to write the corresponding module as a direct sum of modules of the form 𝔸/d𝔸. Write a clear explanation of how this is done for your assignment. A good exposition probably includes some examples.
18 March 2024 Lecture 10: The Krull-Schmidt theorem Hand written Lecture Notes

Student TODO List: Read the notes Krull-Schmidt and torsion. About 70 percent of the questions on the Problem Sheet: Modules are dealt with by the techniques we've developed in the last 3 lectures (i.e. row reduction and changing generators so that the module is a direct sum of A/dA). Discuss with your peers what the efficient to organize and write solutions to these questions is. Write a clear and thorough explanation of how to do this into your assignemt so that whoever reads your assignment has a good guide to how to do any problems of this type that appear on the exam.
19 March 2024 Lecture 11: Jordan Normal form Hand written Lecture Notes

Student TODO List: Read the notes Jordan Normal form. Do some examples of finding the Jordan normal form of a matrix. Explain to a friend how this is done by walking them through how you do it on an example matrix A. Then write an exposition of how to do it into your assignment.
21 March 2024 Lecture 12: PIDs are UFDs Hand written Lecture Notes

Student TODO List: Read the notes PIDs are UFDs Write up a good example showing how the method of proof of the Jordan-Hölder theorem shows that any two prime factorizations of an integer have the same prime factors.

Part C. Rings

25 March 2024 Lecture 13: Euclidean domains are PIDs and PIDs satisfy ACC Hand written Lecture Notes

Student TODO List: Read the notes EDs and PIDs and learn how to do the proofs of the results on this page without referring to notes. Make some further headway on your assignment.
26 March 2024 Lecture 14: Integral domains and Fields of fractions Hand written Lecture Notes

Student TODO List: Read the notes Prime and Maximal ideals and the first page of Fractions and polynomials and do the proofs of these results by applying the Proof machine.
28 March 2024 Lecture 15: Polynomial rings R[x] Hand written Lecture Notes

Student TODO List: Read the notes Fractions and polynomials and do the proofs of the results in the first two pages of this by applying the Proof machine. Start getting your assignment ready for submission on Monday immediately after the break.
8 April 2024 Lecture 16: gcd, lcm, sup, inf, M+N, M∩ N Hand written Lecture Notes

Student TODO List: Read the notes gcds, lcms, sups, infs,P+Q,P∩Q. Careful and helpful peer marking takes longer than you might expect. Do at leaast one peer marking per day (otherwise Sunday night will be painful). Prepare a pdf file for upload for the peer marking component. Make sure that each new marked assignment starts on a new page so that it is easy to separate the pdfs appropriately for return of the feedback.
9 April 2024 Lecture 17: Finiteness conditions and Jordan-Hölder Hand written Lecture Notes

Student TODO List: Read the notes Finiteness conditions and the Jordan-Hölder theorem. Careful and helpful peer marking takes longer than you might expect. Do at leaast one peer marking per day (otherwise Sunday night will be painful). Prepare a pdf file for upload for the peer marking component. Make sure that each new marked assignment starts on a new page so that it is easy to separate the pdfs appropriately for return of the feedback.
10 April 2024 Lecture 18: Principal ideals Hand written Lecture Notes

Student TODO List: Read the notes Principal ideals. Careful and helpful peer marking takes longer than you might expect. Do at leaast one peer marking per day (otherwise Sunday night will be painful). Prepare a pdf file for upload for the peer marking component. Make sure that each new marked assignment starts on a new page so that it is easy to separate the pdfs appropriately for return of the feedback.

Part C2. Some proofs


Proof: Fin. Gen. Modules are sums of cyclics	Proof: Chinese remainder Theorem	Proof: Properties of Tor(M)
Proof: EDs are PIDs	Proof: PIDs satisfy ACC	Proof: Fields have no ideals
Proof: Prime ideals give IDs	Proof: Maximal ideals give fields	Proof: Cancellation law equiv. to no zero divisors
Proof: Correspondence Theorem	Proof: Submodules form a modular lattice	Proof: Finiteness Conditions

Part D. Examples and review for the exam

15 April 2024 Lecture 19: ℤ-modules: review and examples Hand written Lecture Notes

Student TODO List: Do the problems from the problem sheets on ℤ-modules. Be sure that you can do these quickly and smoothly, even under exam stress. Experience shows that there is only one way to achieve this: practice.
16 April 2024 Lecture 20: Jordan normal form and F[x]-modules: review and examples Hand written Lecture Notes

Student TODO List: Do the problems from the problem sheets on 𝔽[x]-modules. Be sure that you can do these quickly and smoothly, even under exam stress. Experience shows that there is only one way to achieve this: practice.
18 April 2024 Lecture 21: Hand written Lecture Notes Mobius transformations and ℙ¹: Automorphisms of ℂ(z)

Student TODO List: Do the problems from the problem sheets on subfields of ℂ(x). Be sure that you can do these quickly and smoothly, even under exam stress. Experience shows that there is only one way to achieve this: practice.
22 April 2024 Lecture 22: Annihilators, Torsion, torsion-free, and free modules Hand written Lecture Notes

Student TODO List: Do the problems from the problem sheets on annihilators, free and torsion free modules. Be sure that you can do these quickly and smoothly, even under exam stress. Experience shows that there is only one way to achieve this: practice.
23 April 2024 Lecture 23: Splitting fields, algebraic, transcendental, separable, normal, perfect, algebraic closure Hand written Lecture Notes

Student TODO List: Do the problems from the problem sheets on algebraic, transcendental, separable and normal questions. Be sure that you can do these quickly and smoothly, even under exam stress. Experience shows that there is only one way to achieve this: practice.


	Proof: Perfect fields give distinct roots

25 April 2024 Lecture 24: ANZAC Day
29 April 2024 Lecture 25: Wedge products, minors and invariant factors Hand written Lecture Notes

Student TODO List: Do the problems from the problem sheets on Smith Normal Form. This lecture tells you how to do the last 3 on this list. Be sure that you can do these quickly and smoothly, even under exam stress. Experience shows that there is only one way to achieve this: practice.
30 April 2024 Lecture 26: Galois Theory and ruler and compass constructions Hand written Lecture Notes

Student TODO List: Do the problems from the problem sheets on Constructible numbers. Be sure that you can do these quickly and smoothly, even under exam stress. Experience shows that there is only one way to achieve this: practice.
2 May 2024 Lecture 27: Gaussian integers, Eisenstein integers and other rings of integers Hand written Lecture Notes

Student TODO List: Do the problems from the problem sheets on Rings of integers. Be sure that you can do these quickly and smoothly, even under exam stress. Experience shows that there is only one way to achieve this: practice.
6 May 2024 Lecture 28: Navigation questions: Writing, induction proofs and groups Hand written Lecture Notes
7 May 2024 Lecture 29: Rational canonical form versus Jordan normal form and groups Hand written Lecture Notes
9 May 2024 Lecture 30: Navigation: Categories (Orbit-Stabilizer, Rank-nullity, M/ker(f) ≃ im(f) Hand written Lecture Notes
13 May 2024 Lecture 31: Splitting fields and making bases and field automorphisms explicit Hand written Lecture Notes
14 May 2024 Lecture 32: Fractional ideals, partial fractions and Dedekind domains Hand written Lecture Notes
16 May 2024 Lecture 33: Rings of integers Hand written Lecture Notes
20 May 2024 Lecture 34: Primitive polynomials and Gauss lemma Hand written Lecture Notes

Proof: Primitive representatives Proof: Primitive polynomial properties
21 May 2024 Lecture 35: Insolvability of the quintic over ℚ Hand written Lecture Notes
23 May 2024 Lecture 36: e is transcendental over ℚ

The last problem session will be on
Friday 14 June from 2-5pm in Russell Love

There will be a problem session 12-2pm on 21 May, but no problem session on Thursday morning 23 May.


Proof: Primitive representatives		Proof: Primitive polynomial properties