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When I am reading a mathematical textbook, I tend to skip most of the exercises. Generally I don't like exercises, particularly artificial ones. Instead, I concentrate on understanding proofs of theorems, propositions, lemmas, etc..

Sometimes I try to prove a theorem before reading the proof. Sometimes I try to find a different proof. Sometimes I try to find an example or a counter-example. Sometimes I try to generalize a theorem. Sometimes I come up with a question and I try to answer it.

I think those are good "exercises" for me.

EDIT What I think is a very good "excercise" is as follows:

(1) Try to prove a theorem before reading the proof.

(2) If you have no idea to prove it, take a look a bit at the proof.

(3) Continue to try to prove it.

(4) When you are stuck, take a look a bit at the proof.

(5) Repeat (3) and (4) until you come up with a proof.

EDIT Another method I recommend rather than doing "homework type" exercises: Try to write a "textbook" on the subject. You don't have to write a real one. I tried to do this on Galois theory. Actually I posted "lecture notes" on Galois theory on an internet mathematics forum. I believe my knowledge and skill on the subject greatly increased.

For example, I found this while I was writing "lecture notes" on Galois theory. I could also prove that any profinite group is a Galois group. This fact was mentioned in Neukirch's algebraic number theory. I found later that Bourbaki had this problem as an exercise. I don't understand its hint, though. Later I found someone wrote a paper on this problem. I made other small "discoveries" during the course. I was planning to write a "lecture note" on Grothendieck's Galois theory. This is an attractive plan, but has not yet been started.

EDIT If you want to have exercises, why not produce them yourself? When you are learning a subject, you naturally come up with questions. Some of these can be good exercises. At least you have the motivation not given by others. It is not homework. For example, I came up with the following question when I was learning algebraic geometry. I found that this was a good problem.

Let $k$ be a field. Let $A$ be a finitely generated commutative algebra over $k$. Let $\mathbb{P}^n = Proj(k[X_0, ... X_n])$. Determine $Hom_k(Spec(A), \mathbb{P}^n)$.

As I wrote, trying to find examples or counter-examples can be good exercises, too. For example, this is a good exercise in the theory of division algebras.

EDIT Let me show you another example of self-exercises. I encountered the following problem when I was writing a "lecture note" on Galois theory.

Let $K$ be a field. Let $K_{sep}$ be a separable algebraic closure of $K$. Let $G$ be the Galois group of $K_{sep}/K$.

Let $A$ be a finite dimensional algebra over $K$. If $A$ is isomorphic to a product of fields each of which is separable over $K$, $A$ is called a finite etale algebra. Let $FinEt(K)$ be the category of finite etale algebra over $K$.

Let $X$ be a finite set. Suppose $G$ acts on $X$ continuously. $X$ is called a finite $G$-set. Let $FinSets(G)$ be the category of finite $G$-sets.

Then $FinEt(K)$ is anti-equivalent to $FinSets(G)$.

This is a zero-dimensional version of the main theorem of Grothendieck's Galois theory. You can find the proof elsewhere, but I recommend you to prove it yourself. It's not difficult and it's a good exercise of Galois theory. Hint: Reduce it to the the case that $A$ is a finite separable extension of $K$ and X is a finite transitive $G$-set.

EDIT If you think this is too broad a question, you are free to add suitable conditions. This is a soft question.

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    doesn't seem like a very useful question2012-06-28
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    The fact that this can vary with the book (including many books, e.g. Lang's *Algebraic Number Theory*, that have no exercises) makes it seem too broad.2012-06-28
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    A few professors have told me on several occasions that it is often not important to understand the proof of a result, but just that it is true. That being said, I agree that this is a broad question, and depends on personal preference. But, I think most people do a combination of what you seem to be doing, but also solve some exercises. One book that comes to mind for which solving exercises seems to be essential to grasp the subject better is Atiyah-Macdonalds's Commutative Algebra. I don't think I would have learnt a lot just by reading the text and skipping exercises.2012-06-28
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    @anthonyquas See my **EDIT**.2012-06-28
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    @Rankeya I skipped most of the exercises of Atiyah-MacDonald. Instead, I read Bourbaki, Matsumura, Zariski-Samuel.2012-06-28
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    It would take me a long while to go through the books you mentioned following your approach. I ideally like to try and prove results without seeing the proof, but I make very little progress this way. Do you progress moderately quickly through books using your approach?2012-06-28
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    @Rankeya You seem to forget that those books have much more coverage than Atiyah-MacDonald.2012-06-28
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    Deciding on just understanding the theory (and constructing the proofs of theorems) but not actually using the theory to attack any problems reminds me of the following apocryphal tale. A (non-athletically-inclined) faculty member read tennis great Bjorn Borg's comment that tennis was a very simple game; all that needed to be done was to hit the ball over the net one more time than the other guy. He felt he understood the theory perfectly and so promptly entered the tennis tournament at the faculty club. He could not understand why he lost in the first round without winning even one point!2012-06-28
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    @Dilip You will use the theory sooner or later, otherwise the theory is not for you. For example, when you learn commutative algebra, you have to use group theory, field theory, Galois theory, etc.. When you learn algebraic geometry, you have to use commutative algebra, homological algebra, algebraic topology, complex analysis, differential geometry, etc..2012-06-28
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    @Rankeya You wrote "Do you progress moderately quickly through books using your approach?" No, but I think it's one of the most effective ways to learn a subject.2012-06-29
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    @DilipSarwate, please make your comment an answer. I want to upvote it.2012-06-29
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    Also, in my personal experience (not that I have a lot), the difference between reading a textbook and reading a textbook and doing the exercises is the difference between skimming a book and reading a book. It is like having severe astigmatism versus seeing perfectly clearly. It can also be so bad that you don't even realize it. (I've read a chapter and then tried the exercises, only to find that I completely did not understand the chapter.)2012-06-29
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    @Dilip You seem to misunderstand my methods. My methods demand *creative* efforts.2012-06-29
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    @Limitless You seem to misunderstand my methods, too. Please read all the **EDITs**.2012-06-29
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    @MakotoKato, sorry, but you misunderstand. I am not remarking on your methods what-so-ever. In fact, I was remarking entirely on Dilip's opinion. Your methods actually sound good. But I am agreeing with Dilip in the sense that if you _only_ read a textbook, things will be exactly as he has stated. Are we understanding one another?2012-06-29

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