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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.
Attempted recording of these sessions is as follows:
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
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
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.
- 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
- 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.