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I believe that the proof in Marcus's "Number Fields" contains a proof which does not rely on integral closure except to say that if an element satisfies a monic polynomial, it is in the domain. I'll summarize the lemmas he uses:

1) For any ideal $I$, there is an ideal $J$ so that $IJ$ is principle.

2) For any proper ideal $I$, there is an element $x$ in the field of fractions and not in the Dedekind domain so that $xI$ is still in the Dedekind domain.

To prove the first second lemma he uses integrally closed, but only to show that an element of the field of fractions satisfies a monic polynomial, and is thus in the Dedekind domain.

3) The ideal classes form a group. This is a quick consequence of the previous lemmas.

4) Some group results about the ideals. (the google books view which I am using is missing the last page).

I think after that there is no more use of integrally closed, but as I said I'm missing the last page of the proof. Hope this helps.

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I believe that the proof in Marcus's "Number Fields" contains a proof which does not rely on integral closure except to say that if an element satisfies a monic polynomial, it is in the domain. I'll summarize the lemmas he uses:

1) For any ideal $I$, there is an ideal $J$ so that $IJ$ is principle.

2) For any proper ideal $I$, there is an element $x$ in the field of fractions and not in the Dedekind domain so that $xI$ is still in the Dedekind domain.

To prove the first lemma he uses integrally closed, but only to show that an element of the field of fractions satisfies a monic polynomial, and is thus in the Dedekind domain.

3) The ideal classes form a group. This is a quick consequence of the previous lemmas.

4) Some group results about the ideals. (the google books view which I am using is missing the last page).

I think after that there is no more use of integrally closed, but as I said I'm missing the last page of the proof. Hope this helps.