It seems that the ratio of those authors allowing a field to be a Dedekind domain to those who do not is almost 50 - 50. Why such a bewildering lack of consensus for such an elementary notion?
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$\begingroup$ Perhaps because for any interesting fact about Dedekind domains the potential for confusion is never noticed since whatever is being said is obvious in the field case (or at least very well-known to anyone first learning about Dedekind domains). I doubt at the post-textbook stage of math that anyone would regard fields as Dedekind domains. (I am counting Bourbaki as a textbook, by the way.) A good analogy is whether a differential geometry book implicitly assumes manifolds have positive dimension or not: probably goes either way, depending on the source. $\endgroup$– BoyarskyCommented Jun 20, 2010 at 3:53
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4$\begingroup$ If you think of a Dedekind domain as a regular (or normal) one-dimensional Noetherian domain, then you know that fields are not allowed. If you think of a Dedekind domain as a domain for which every fractional ideal is invertible, or as a domain in which the monoid of nonzero ideals has a basis, then you know that fields are allowed. Esthetically, both points of view make sense to me. If you adopt the second one then you will have to state trivial exceptions to some theorems. $\endgroup$– Tom GoodwillieCommented Jun 20, 2010 at 4:50
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3$\begingroup$ Are you looking for a mathematical or a sociological explanation? $\endgroup$– S. Carnahan ♦Commented Jun 20, 2010 at 5:20
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$\begingroup$ There are a lot of elementary definitions in mathematics for which there is no consensus. This especially happens when the differences in the definition are so small as to make no real difference in the resulting theory: it then becomes mostly a matter of convention and aesthetic... $\endgroup$– Pete L. ClarkCommented Jun 20, 2010 at 7:39
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1$\begingroup$ @Pete: Thank you for the information about format. To your other point, I acknowledge that creating a tag for the empty set bordered on frivolity, but I do think that if one wanted a tag for the general topic of "taking trivial cases of definitions seriously" then empty-set is as good a short phrase as any. $\endgroup$– Tom GoodwillieCommented Jun 20, 2010 at 11:21
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Historically Emmy Noether's paper introducing the concept "Dedekind domain" certainly included fields (see e.g. Kleiner's book on the history of abstract algebra, which gives axioms). She was characterising the scope of unique factorisation into prime ideals. Now, a question that might actually be answered is "what happened after the late 1920s in commutative algebra to change this?" To which there is a fairly clear answer, implied by Goodwillie's comment: geometric concepts now mean more than those derived from algebraic number theory. "Dimension 1" seems more a propos or right-thinking than "dimension at most 1".
answered Jun 20, 2010 at 13:25
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1$\begingroup$ Charles, that's almost exactly what I was getting at. Yet when I think of calling a field a Dedekind domain I am not completely giving up the geometric viewpoint; I am thinking of it not as a $0$-dimensional object but (informally) as a $1$-dimensional object from which all of the $0$-dimensional points have been removed. $\endgroup$ Commented Jun 20, 2010 at 19:27