@Harry. Unlike you I had given up trying to read Bourbaki, and I also gave up logic, set theory etc.. So I must confess I am trying to argue based on gut feelings.

@Harry. A proof by contrapositive is still a "direct proof". The second type of proof above, is not. It is true reductio ad absurdum.

@Harry. Yes that is true. But there are genuine examples of ad absurdum, like the Cantor proof.

Ah, F G Dorais has already written it below. I had not seen it yet when I was typing the above comment. Sorry.

This works only for "P implies Q" types of statements. What do you have to say about "P is not true" type of statements? Here specifically I have the Cantor's proof that a set and its power set are not equivalent.

Maybe this is not specific to moduli space. But I did not know about the things you write, and I learned something from your answer. Thanks.

@Ryan. Your deleted answer was helpful.

I know of Moduli spaces as $\mathcal{M}_{g,n}$, the moduli of curves with n marked points,from algebraic geometry. So I am familiar with the algebraic geometry approach. I want to grasp the flavor and purpose of the Teichmuller theory approach, now that I have understood the first principles of the connection with Teichmuller space with and mapping class group.

And fractional ideals are invertible anyway, since they are finitely generated.

It is proved in Jacobson, Basic Algebra, Vol 2., that unique decomposition into prime ideals is equivalent to being a dedekind domain.

I am able to prove this statement itself in a very straightforward manner using axiomatic properties of the determinant. I am only worrying about the more general theorem.

Don't get hurt that you got a neg vote. It happens sometimes. I was not the negvoter; but I suppose you would not have got it if you had included the exercise in your question. Make questions clear. At least now if you can edit and include the exercise, it would be nice.

Not all of us have the book at hand. Can you give more information on the question?

@Emerton. Can we recover a number field from the knowledge of Artin motives attached to it? In general, is it possible to recover an algebraic variety from a motivic understanding, i. e., is it possible to get back to the original thing, if you know the motivic chunks as you called them?