website for
Galois Theory
(2010, M24)

This is a website for the Part IID course Galois Theory (2010, M24); here are some material for this course.

(29/11/10) Thank you so much for your sincere (and often heartwarming, um, seriously) feedbacks in the questionnaires. I really appreciate. I may try to answer some of them (mainly mathematical ones) on this page (I did - 30/11/10). Also I'll occasionally update the notes when I have time.
(Common sense) comments on exams. Only the intersection of the schedule and what I did in the lectures are examinable. If you prefer to use different definitions/proofs from the lectures, it is totally acceptable as long as they are clearly stated and the essential arguments are given (although the questions will be designed to have as little such complication as possible).

(from the official website of the department)

Meetings: Mon / Wed / Fri 11am, at MR 3.
First lecture: Fri., 8th October, 2010.
Last lecture: Mon., 29th November. There will be no lecture on Wed., 1st December!

  • p.7 of the [pdf] Courses in Part II of the Tripos,
  • p.29 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.


    I will have two versions of lecture notes posted here. There is a 20+ pages appendix of linear algebra and groups/rings/modules to make them self-contained - it may present the things you already know in a different language. These files are tentative and will be constantly updated - always check the latest file!

  • [pdf] Short Notes (last update 24/11/10)

  • [pdf] Long Notes (last update 24/11/10)

    The Short Notes are basically the transcription of what goes on the board (1 page per lecture), and logically complete. This is the kind of writing you need to read and decipher as you study maths further through literature. The Long Notes come with explanations and complements of all sorts (but still 2 pages per lecture).

  • [pdf] Minimal Notes (last update 26/11/10)

    The Minimal Notes is for the experienced: I attempted to give a shortest possible proof of the Galois Theory and the impossibility of solving quintics by radicals (building on Dedekind/Artin theorems, thus based on the approach of Lecture 21 and not the main lectures). It should contain all the essential arguments; comments welcome!

    ______example sheets

    The problems are mostly from the past sheets, with different order.

  • [pdf] Example sheet 1 (Lecture 1-6, handed out 13/10/10, last update 20/10/10)
  • [pdf] Example sheet 2 (Lecture 7-12, handed out 27/10/10, last update 5/11/10)
  • [pdf] Example sheet 3 (Lecture 13-18, handed out 10/11/10)
  • [pdf] Example sheet 4 (Lecture 19-23, handed out 24/11/10)

    Comments on the optional questions. I chose eleven questions from each sheet that you might try first (and these should suffice for the sake of preparing for the exam). I decided to call the rest "optional" (the starred ones). The easy ones are always unstarred and the very hard ones are always starred, but the division is not only in terms of the difficulty; unstarred doesn't mean easy. Not only in terms of importance either; there are interesting and educational questions among the starred. I definitely expect the good students to find interesting questions among the starred ones to challenge themselves.

    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.


    What I think about my lectures (30/11/10). Overall I did my best (during the term all my time and effort was put into this course), although my self-evaluation would be somewhere around 3-3.5 out of 5. What I think I definitely should have done: more discussions on writing down Roots and Homs explicitly, with some examples. All the proofs I did are logically equivalent to the standard ones in the literature, but I certainly tried to put them in a framework which is compatible with how I understand the subject at the research level. This doesn't mean I had to assume more knowledge (after all, they are just Sets and Maps), but it could have been that it seemed to require more training in abstract thinking. And although I didn't omit any of the proofs, it seems that the way I write proofs look unfamiliar or inadequate - I am only starting to see what's the problem, but I wish there were more chances to discuss particular proofs with students personally in detail, to figure this out.

    Here are some answers to the comments from students. This is _not_ an attempt to represent my student evaluation, nor to justify myself. If your comments don't appear or seemed to be classified wrongly, don't worry; I'm not taking any offense, just grouped them for convenience in answering them. Thank you so much for your positive or constructive comments!

    I had absolutely no idea what on earth "Hom-counting Lemma" was meant to mean. Sorry. It is Lemma 13.3 of the printed notes, the one with 5 parts. Do ask me or email me right away when you have such questions!
    We seemed to prove many theorems in two different ways. Sorry if I confused you. We did prove the fundamental theorem in two different ways, but it's not because I changed my mind, it was planned. I hope looking at the "Logical Order" diagram in the beginning of the notes helps.
    Would it be better if the syllabus goes like: Equations -> field extensions -> splitting fields -> separable ext -> normal ext -> FTGT -> radical -> soluble instead of going into FTGT straight away? There are many great books along those lines, I believe. Some reasons behind my approach: (1) I see FTGT as something logically independent from all else (see the Minimal notes), so it's good to make this fact clear. (2) I wanted to show you the examples of Galois correspondence for cyclotomic extensions and finite fields, which are all simple extensions, before confusing you with separability.
    Could have discussed inseparable extensions in more detail. Yes, but inseparable extensions do not fit into the main objectives of this course, and I personlly think they are understood better in terms of algebraic geometry in positive characteristic, so I treated them as anomalies. That's why I didn't treat normal extension, which is simply Galois extension when it's separable, and we don't look at "normal but inseparable" ones. (I don't mean to be dogmatic at all.)
    Trace and norm are in the schedules - not lectured? I didn't. The connection with the main part of the course was tenuous. I will include them in the printed notes but will not be examinable. Sorry for violating the "schedule is minimal for lecturing..." principle.
    When explaining extensions of K, stress that there is an isomorphic copy of K in the extension field. Yes, thanks. I think I said it twice but cannot be too much stressed. It's one of the crucial difference between the abstract and concrete approaches. This definition makes the proofs simpler, while sometimes could be confusing when you try to imagine everything inside a fixed extension. I tried to steer clear of this confusion in all the proofs I made, but could work more on this. (This definition is adopted in Bourbaki, Algebre 5.2.1 Def 1.)
    Took me a while to realise that determining the image of X bar in K_P determines the Hom, perhaps explain better. Agree. I later got concerned that this bit deserved more explanation, and it might have tripped up many students.
    The linear-algebraic proofs were easier to understand so perhaps should be presented first. That is a possibility - I found it rather less intuitive than the proof for simple extensions (Lecture 6), but it requires less. See the Minimal Notes.
    The comments at the end linking the concrete and abstract approaches were interesting. Perhaps there should be a Part III course focusing on the abstract approach? I thought about it too, and it is a possibility. What we need to introduce is the tensor products of algebras, and that's pretty much enough to do everything canonically. Assuming representation theory of finite groups, we can directly formulate Galois theory as the categorical equivalence between Artin motives and Galois repesentations (finite Artin representations).

    Include more proof details into online notes. / Please err on the side of writing too much. Yes. Perhaps there's something about my own mathematical background, that tells me that having redundant words in proofs will make them HARDER to understand - contrary to "err on the side of writing too much" (good for exams perhaps). But sometimes I do find other peoples' proof hard to follow for lack of details. As I think all mathematical inputs are written, I really need to know what you mean by details - it has nothing to do with being clever/dumb, it's linguistics.
    The written babble in the long notes is a very good idea / I really liked the philosophical comments/motivations etc you had in the long notes - why did they later disappear? Thanks. I am sorry I couldn't continue writing long notes. After Lecture 9, sorting out the proofs for the short notes to make it clearer took up all of my time. I will try to fill them in, so check them again before the exam.
    When proving propositions with multiple parts in your notes after proving (i)->(ii) start a new line to prove (ii)->(iii). / They would be better if the lecturer didn't try to fit one lecture on a single page. I know this is idiosyncratic or even idiotic; I just like to see everything in one page at once, but I never omitted essential details. Forgive me!
    I very much appreciate the succint nature of the online notes. It's good to know that someone shares my taste.

    ___Example Sheets.
    I like that there are lot of Qs to revise from. / Setting optional questions is a really good idea on the example sheets. Good. This was a major problem last year when I didn't put the stars.
    Some questions were interesting - others were decoupled from the course. / Also, I wondered why the lecturer kept saying he found the sheets horrible/hard etc... surely he could have changed them? Yes. Well, I didn't have time yet - I only checked that they are all solvable using the things I did in the course, and that was already "horrifying" me! I do admire this set of questions that were accumulated in Cambridge over a long period of time, and as a show of respect I didn't feel like changing them too much. But perhaps I should start changing them.

    (1) Write in full sentences. (2) Write down all important spoken comments. (3) Write stuff in order i.e. don't go back to a previous board to write a remark. I definitely am aware of the difficulty of taking notes. The best way to improve on all 3 points would be to prepare a complete handwritten notes and copy them on the board, which I sometimes did - but it could result in less inspiring/fun lectures (for myself at least). I'm not skilled enough to be spontaneous and unerring at the same time. I think I have to make a lot of effort to make my boardwork satisfactory. About going back to a previous board, could it be better if I used yellow chalks for added/corrected parts?
    I actually enjoy your comparatively disorganized lecture style and in particular your intuitive discussions. / I don't like Cambridge system of lecturing that much. / The lectures are not in Cambridge style. There is a lot to be said about this. I don't mean to assert my own style against the system so I do try to improve my boardwork, and meanwhile my printed notes are there to help you. I do appreciate the good things about "Cambridge style", in which students can claim the ownership of the matrial through an immaculate note-taking. And that results in pleasure/enjoyment as well. But some of you do not need this and have other ways to learn, and I want to keep some spontaneity. Anyway, the bottom line is that, even if there is a Cambridge style lecturing/learning, there is no local style for mathematics itself - conveying/understanding mathematical proofs is truly universal, and that's what's wonderful about it. To make my language more familiar to you (connecting to Part IA/IB maths) is another matter, where I should work hard.
    So often the most important things (in terms of understanding the proof) you just said and did not write down! / His board work was strings of symbols mostly, even putting words like 'so', 'but', 'if' etc would have been VERY helpful. Thanks! I'll try. I came from Japan where we frantically tried to jot down the things the lecturer said. Also I always thought writing more words in proofs made them harder to read. In America students will stop me whenever there's something they cannot follow, which informed me what kind of details are needed. Now I should learn different perspectives.
    Arguments of the form (ii)->(iv): take L=K, E=F use previous Lemma (i)<-(iii) is impossible to follow in class. / The proofs should be more self-contained without so many references to other numbered results. / Some of the proofs were quite difficult and took several hours to figure out, e.g. the ones in notes for Sep. Ext. I. That happend in Lectures 13-15. Maybe I should have done better in sorting out the statements. Also, I think I showed you a style of stating/proving things in the way you weren't familiar with. Great if you managed to figure out by yourself - it would have been nice if you could email me back then, so that I could see what kinds of things could be difficult for the students. On the other hand, when the proofs look very succinct on the board, it usually meant that it's literally just plugging in L=K and E=F, if you didn't mind flipping over 2 pages of your notes to see the Lemma from previous lectures, so I thought it's not worth writing the statements again. But it could be very annoying, I agree. About building on previous results - I feel more insecure when we don't build up on results from previous lectures, in which case the material start to seem like juxtapositions in an arbitrary order. In general, I think in Part II things start building up vertically, much more than in Part IA/IB.
    It is good for the students who already know Galois theory. / Maybe he is just unsure about how much we know - in that case maybe he should assume less. The truth is that I don't assume any knowledge on your side, so I suspect the problem is about how we think about proofs and examples. I'm currently learning a lot (through giving supervisions etc) regarding what could be the problems.
    I love your stories, comparisons, way of expressing yourself. I used to write all the jokes down. Ah, my jokes should be kept between you and me!

    ___Further comments (better if you point to specific mathematical points) always welcome.

    ______old stuff

  • 2009 course website

  • Last modified: November 30, 2010.