Galois Theory (2009, M24)

This is a website for the Part IID course Galois Theory (2009, M24); here are some material for this course.

------ time

Meetings: Tue / Thu / Sat 11am, at MR 2.
First lecture: Thu., 8th October, 2009.

The lectures on Tue., 24th and Thu., 26th, Nov., 2009 will be by Prof. Tony Scholl.

------ syllabus (from the official website of the department)

  • p.7 of the [pdf] Courses in Part II of the Tripos,
  • p.27 of the [pdf] Schedules of the Tripos (I will lecture in a different order; see the notes below),
  • past example sheets on Galois Theory page.

    ------ notes

    I will have two versions of lecture notes posted here. I attached 40 pages appendix of linear algebra and groups/rings/modules to make them self-contained. These files are tentative and will be updated as the lectures are given - always check the latest file! (But I'll try not to change the notes for the finished lectures unless there are serious mistakes.) (Well now I have to retreat this comment; I do change things in retrospect, not essentially but in terms of notation and orderings...)

  • [pdf] Short Notes (last update 28/11/09)

    The Short Notes are the bare minimum of mathematical content (1 page per lecture), nonetheless logically complete. This is the kind of writing you need to read and decipher as you study maths further through literature.

  • [pdf] Long Notes (last update 28/11/09)

    The Long Notes come with explanations and complements of all sorts (but still 2 pages per lecture). Lectures will follow these notes.

    ------ example sheets

    The problems are from the past sheets, with different order.
    Dear supervisors: shoot me an email! --- let me send you the solutions for the example sheet that I prepared (but don't show them to the students!). Hope you can give me feedbacks.

  • [pdf] Example sheet 1 (Lecture 1--6)

  • [pdf] Example sheet 2 (Lecture 7--12)

  • [pdf] Example sheet 3 (Lecture 13--18)

  • [pdf] Example sheet 4 (Lecture 19--23)

    ------ comments

    22/11/09 (in Tokyo) Wrote some notes for Lectures 20 onwards.

    20/11/09 (evening) Wrote the notes for Lectures 16 and 18, updated 17 and 19 as well.

    16/11/09 (evening) Finalized the Example sheet 3.

    14/11/09 Finally typed up the notes of Lectures 7 and 12 --- sorry for the long delay! I'll work on problems and finalize Ex 3 soon.

    9/11/09 (evening) Upon urgent request from some of you, I prepared the tentative version of Examples sheet 3 (again I expect this to be very close to the final version). I hope this helps!

    9/11/09 I didn't have to change the Example Sheet 2 too much --- I took away Problems 3 and 5, because Problem 3 might look more familiar after doing Kummer Theory, and Problem 5 deals with the transcendental simple extension (which I didn't really define; in my notes simple extensions are always finite) and is essentially Problem 10(ii) of Example Sheet 1 anyway. I fixed some typos, changed some notation, added a trivial theoretical piece 12.

    8/11/09 (evening) Now I seem to have finished sorting out the arguments in Lectures 13-15; I hope the obscure/redundant arguments in the older version has been cleared. Lecture 6 has been revised too (in particular, check out the new notation F_\tau for (F,\tau).) Now I'll work on the example sheets as well!

    5/11/09 There seems to have been some concerns about the timing of the example sheets --- I was told to give example sheets at the end of 2nd, 4th, 6th and 8th weeks, but apparently it's more convenient for the supervision to work if I give them out at the end of 1st, 3rd, 5th and 7th weeks. Sorry about that! I'll try to work faster on the example sheets but I have to stick to my original schedule this year, partly because I am organizing the course as the lectures proceed. Meanwhile I put up the tentative version of the 2nd sheet --- perhaps some of the questions need facts from later lectures, but most of them should be fine. Will finalize at the end of this week.

    22/10/09 --- successfully back from Chicago and back to work; the example sheet 1 is on the web (the hard copy I distributed had a trivial typo).

    13/10/09 End of the first week --- thanks for coming to the lectures. As I promised, I deliberately made the first three lectures less structured so that I can stimulate your curiosity, and also to convey the spirit of Galois theory, which may seem quite inscrutable if you just follow the mathematical statements in the way you learned Linear Algebra or Analysis. There is a hidden mystery, highly counter-intuitive, behind the seemingly innocuous, intuitive objects like polynomials and their roots, and we need to change our language/viewpoint in order to get our hands on the mystery. And we need to understand the motivation and need for changing into more abstract, theoretical and modern language, especially if you plan to study further than Galois theory.

    If my lecture style is starting to make you feel worried, don't worry too much, I'll make it closer to other people's lectures from Lecture 4. On the other hand, the logical structure of Definitions, Propositions and Proofs are spelled out in the Short Notes, so if your own notes look too disorganized because of my lecture style, you can always rely on the notes on the website --- I don't see much point in reproducing and copying down exactly the same thing as I put on the web!

    As for the example sheets, I will rearrange the order of the past questions. For example, Questions 2,3,4,6,7,8,9,12,14 of Example sheet 1 of 2006 should be more or less accessible at this point.

    maintained by Teruyoshi Yoshida
    Last modified: Nov. 28, 2009.