MATH 338: ALGEBRA IIIB; 2004 D2
STUDY GUIDE

LECTURERS

GALOIS THEORY	Dr Gerry Myerson	E7A 415	98508952	gerry@math.mq.edu.au
RING THEORY	Professor Ross Street	E7A 311	98508921	street@math.mq.edu.au

LECTURES

There are four hours of lectures per week:

Monday	11am	E6A109	Street
Tuesday	11am	W5C312	Myerson
Wednesday	11am	W5C234	Street
Friday	11am	W5A204	Myerson

2003 Examination: Don't get too excited when you see 2004 on the pdf file; it is an idiosyncracy of the TeX program that "corrected" the date.

TEXTBOOKS AND REFERENCES

• Ian Stewart, "Galois Theory" (Chapman and Hall, 2nd Edition 1989 which is what the lecturers have, or 3rd Edition)
• John A. Beachy, "Introductory Lectures on Rings and Modules" (Cambridge 1999)
  Here are some reference books on Galois Theory with a few comments on each:
  • Harold M. Edwards, "Galois Theory" (Springer, 1984, Graduate Texts in Math. 101) [written in the spirit of 'Read the masters!', there is a definite attempt to expose Galois' original ideas]
  • Emil Artin, "Galois Theory" (Notre Dame Mathematical Lectures 2 , 1959) [the pithy work of a master - very thin]
  • Francis Borceux and George Janelidze, "Galois Theories" (Cambridge Studies in Advanced Math. 72, 2001) [the early sections are appropriate for this unit; the keen student can then find how Galois' ideas have developed in recent times]
  • Tom Petsinis, "The French Mathematician, A Novel" (Penguin, 1997) [non-technical novel written in the first person as Galois, sets the historical stage for Galois' work; a fun read of a sad tale!]

    Here are some books with Galois theory, ring theory and more:

  • Serge Lang, Algebra (Addison Wesley, 1971)
  • Garrett Birkhoff and Saunders Mac Lane, A Survey of Modern Algebra (Macmillan, 4th Edition 1977)
  • Saunders Mac Lane and Garrett Birkhoff, Algebra (Macmillan, 1971)
  • Nathan Jacobson, Basic Algebra I (Freeman)
  • Oscar Zariski and Pierre Samuel, Commutative Algebra, Volume I (Van Nostrand, 1958)

  ASSESSMENT

  S1 + S2 + S3 10%; M1 + M2 6%, MP 14%; Galois Theory Final Exam 30%; Ring Theory Final Exam 40%

  WEB SITE

  The homepage for this unit is <http://www.math.mq.edu.au/undergraduate/math338.html>.

  SYLLABUS

  Galois Theory:

  Chapters 1 through 4 and 7 through 14 of Stewart:

• Fields, polynomials, the Division Theorem, Euclid's algorithm.
• Factorization of polynomials, tests for irreducibility.
• Field extensions, isomorphism of extensions, degree of an extension.
• Groups of field automorphisms, fixed field of an automorphism.
• Normal extensions, separable extensions.
• The Galois correspondence.
• Review of solvable groups.
• Solving equations by radicals, the unsolvability of the quintic.
• Other topics, if time permits.

  Ring Theory:

  Chapters 1 through 3 of Beachy, excluding the starred sections:

• Elementary properties of rings, ring homomorphisms, localization of integral domains, unique factorization.
• Modules: direct sums and products, semisimple modules, chain conditions, modules with finite length, tensor products, modules over principal ideal domains.
• Structure of non-commutative rings: prime and primitive ideals, the Jacobson radical, semisimple Artinian rings.

  ASSIGNMENTS

  There will be three assignments (S1, S2 and S3) on Ring Theory, two (M1 and M2) on Galois Theory, and a project MP on Galois Theory.

  Assignments must be submitted by 6pm on the due date as follows:

  Assignment	Week	Due Date	Solutions
  M1	3	Wednesday 18 August	Available
  S1	5	Wednesday 1 September	Available
  M2	7	Wednesday 15 September	Available
  S2	9	Wednesday 13 October	Available
  MP	11	Friday 5 November	Not yet
  S3	12	Wednesday 3 November	Available

  You may put your assignments in the relevant assignment box or hand it to the Lecturer after the Wednesday class on the due date. The boxes are located in the corridor between E7A and E7B, near Room E7B 363. You must put your name and student number clearly at the top on the front page. Pages should be securely stapled together, and folder and plastic sleeves must not be used. Poorly presented work will be given the mark of zero (0). Late assignments will not be marked; where there is documented medical or other evidence that it has not been possible to submit in time, a mark consistent with your work on other assignments will be used.

  Warm-up Exercises for Ring Theory: pdf file

  PRIZES AND SCHOLARSHIPS

  The Frederick Chong prize, valued at $250, is awarded annually for proficiency in 300-level mathematics.

  http://www.maths.mq.edu.au/fchong/

  http://www.cal.mq.edu.au/2001/p3/pt3d_109.htm#The_Frederick_Chong_Mathematics_Prize

  A Doris Wallent Scholarship, valued at $3000, is available to a student with a GPA at least 3.25 in 300-level mathematics who will be able to complete mathematics honours by the end of that year. Under the terms of the bequest preference will be given to a suitably qualified female student.

  http://www.cal.mq.edu.au/2001/p3/pt3d_108.htm#Doris_Wallent_Scholarships_in_Mathematics

  Macquarie University Innovation Scholarships

  http://www.cal.mq.edu.au/2001/p3/pt3d_108.htm#Macquarie_University_Innovation_Scholarships

   

  EXPECTED PARTICIPATION

  Students are expected to demonstrate their participation in this unit by satisfying the following:

  a reasonable attempt on the project and on at least four of the five assignments.

  Please contact one of the lecturers in case of difficulty over this.

  Withdrawal

  Your attention is drawn to the following deadline:

  Tuesday 31 August: Last day for students to lodge a "Change of Program" form to discontinue the unit and to be regarded as not effectively enrolled (NE). See page 89 of the 2004 Macquarie University Handbook of Undergraduate Studies.

  If you submit a form to withdraw after 31 August, you will be awarded the grade FW. A grade of W (withdrawal without academic penalty) will only be awarded in the case of fully documented "unavoidable disruption". See pages 36, 37, 38 and 89 of the 2004 Macquarie University Handbook of Undergraduate Studies.

  Plagiarism

  Plagiarism involves using the work of another person and presenting it as one's own. Any of the following acts constitutes plagiarism unless the source of each quotation or piece of borrowed material is clearly acknowledged:

  • copying out part(s) of any document or audio-visual material (including computer based material);
  • using or extracting another person's concepts, experimental results, or conclusions;
  • summarizing another person's work;
  • in an assignment where there was collaborative preparatory work, submitting substantially the same final version of any material as another student.

  Encouraging or assisting another person to commit plagiarism is a form of improper collusion and may attract the same penalties which apply to plagiarism.

  Any student suspected of plagiarism may be asked for an explanation in the first instance. If no satisfactory explanation is forthcoming, the student may be penalized by being awarded negative full marks for the assignment(s) concerned. Any further infringement will be reported to the University for consideration by the Disciplinary Committee.

  Please see the section concerning plagiarism on pages 42-43 in the 2004 Handbook of Undergraduate Studies.

  Special Consideration

  If illness or misadventure make it impossible for you to sit for the final examination, or interfere significantly with your performance in the examination, you are permitted to request "special consideration". If we are satisfied (eg, by your previous performance in the unit, or by the quality of work you have been able to produce in the examination) that there is evidence that you have not been able to show your true ability, we may decide to invite you to sit for a special examination to resolve your grade for the unit.

  Notes

  • Special consideration will only be given to students who have attempted at least four of the five assignments and who have attempted both the written tests. Please contact one of the lecturers in case of difficulty over any of these.
  • It is the Division's decision whether to offer a special examination or not. Students cannot apply for one, or expect to be granted one automatically, and requests "to sit for the special exam" will not be looked upon with favour.
  • It is essential that you notify the Registrar in writing of the misadventure, accompanied by any appropriate documentary evidence. (See pages 40 & 89 of the 2004 Macquarie University Handbook of Undergraduate Studies.) It is also advisable for you to let us know informally that you have applied to the Registrar.
  • The purpose of the special examination is to resolve the temporary difficulty caused by your illness or misadventure; the purpose is NOT to give you an advantage over other students by allowing you extra time to study. For this reason we will hold the special examination as soon as possible, and in determining your grade for a special examination we may take into account the possibility of extra study time available to you.
  • In view of the previous paragraph, you must make sure you are readily contactable, and must hold yourself available to sit for the exam at short notice on the date and time we set. Address information provided to the Registrar must be up to date.
  • It is expected that if there is a Special Examination then it will be held on Friday 17 December at 9.20am in room E7B 264.
  • If you elect to be away from Sydney during the week of the special examination, and so cannot be contacted or are unavailable to sit for the examination, we will accept this as a firm indication that any request you may have submitted for special consideration does not apply to this unit.

  Learning Outcomes and Grades

  The outcome of this unit is an understanding of the various topics that are presented.

  The level of understanding is reflected in the grade awarded.

  For an explanation of the grades, please see Rule 10 of Bachelor Degree Rules, concerning grades, on pages 88-89 in the 2004 Handbook of Undergraduate Studies.

  Generic Skills

  The Macquarie experience is designed to lead students to a career in the city and a place in the world. It encourages life-long learning and links teaching to cutting-edge research.

  Macquarie seeks to develop generic skills for students, building flexible outcomes for life and for the workplace over a life's career. These skills include:

  • foundation skills of literacy, numeracy and information technology;
  • self-awareness and interpersonal skills, such as the capacity for self-management, collaboration and leadership;
  • communication skills for effective presentation and cultural understanding;
  • critical analysis skills to evaluate, synthesise and judge;
  • problem-solving skills to apply and adapt knowledge to the real world; and
  • creative thinking skills to imagine, invent and discover.

  Mathematics is an ideal medium to learn problem solving skills. Many problems occurring in the real world can be described as mathematical problems, and are then investigated thoroughly using all the mathematical techniques available. The solutions are then interpreted in suitable ways to provide answers to the original problems. In this unit, you will be trained in a number of these mathematical techniques, as well as some of the ways to apply them.

Survey

A small questionnaire concerning this unit is on the web. Please take a few moments during the last week of the semester to answer the questions and add any comments you deem appropriate.

The address is http://rutherglen.ics.mq.edu.au/~macqtex/feedback.html