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I guess not. However, I think I have to understand proofs of some of them, if not all of them. So what is the criterion, if any? What kind of theorem whose proof I can get away with?

EDIT For example, let me take the main theorems of class field theory. It's rather easy to understand what they say, but it's difficult to understand the proofs. Do I have to understand those proofs to research on algebraic number theory?

EDIT Another example: Hironaka's theorem on resolution of singularities of an algebraic variety in characteristic 0. I guess most algebraic geometers can do their research without understanding its proof.

EDIT I'll add more examples.

  • The theorem that singular homology groups satisfy the Eilenberg-Steenrod axioms.

  • Most of the basic theorems of homological algebra, for example, the theorem that a filtered complex has a spectral sequence.

  • The Feit-Thompson theorem that every finite group of odd order is solvable.

  • The classification of finite simple groups(CFSG).

EDIT I believe this question is becoming more and more important because mathematic is developing faster and faster than before. I guess you are going to give up understaning the proofs of all the important theorems in your field, regardless whether you want to understand them or not.

EDIT The existence of the field of real numbers. I'm almost certain that one doesn't need to know its proof to do analysis. All one needs to know are its properties.

  • 3
    This question is either too broad or too soft. Do you have any concrete example in mind?2012-05-29
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    This question is far too broad. What field? What counts as an important theorem in that field?2012-05-29
  • 0
    Depends on what you are trying to do.2012-05-29
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    I think the question is apllicable to any field, but if you want an example, algebraic number theory as described in Lang's or Neukirch's book.2012-05-29
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    I suggest you first read and understand what the theorems say and how they hang together. As an exercise, can you do that from memory for elementary group theory and elementary ring theory? Or linear algebra for that matter. I mean, just list the theorems in a logical order.2012-05-29
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    I think Matt E.'s answer [here](http://math.stackexchange.com/a/149310/5363) is relevant to this question.2012-05-29
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    [This question](http://math.stackexchange.com/questions/118158/do-you-prove-all-theorems-whilst-studying/118275#118275) seems related.2012-05-29
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    CFSG... no single person understands that proof. Conceivably, no single pair does either. And AFAIK, no single person would want to.2012-05-30
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    @mixedmath: I'm not sure what exactly you mean by "would want to", but I would want to all right, if I weren't so concerned with effectively spending my time on broad number of fields.2012-05-30
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    @Lovre: I refer to the fact that the general proof is tens of thousands of pages long, and has not been verified by any human. I don't want to spend my time on it, even though I *do* use it from time to time and it's in an adjacent field.2012-05-30
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    Admittedly the question is vague but apparently was sufficiently clear to garner an answer that received 10+ upvotes; as such I don't think it deserves to be closed.2012-05-30
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    The question is precise and the examples are extremely well-chosen.What is being asked is extremely clear and the OP has an excellent conjecture on the answer. Whoever thinks that "This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form" should wonder how come Matt answers it so well.2012-05-30
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    @Georges Couldn't have said it better, casting the final vote to reopen.2012-05-30
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    Thanks for the kind words, @Alex, and more importantly for your vote.2012-05-30

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