Definition talk:Field of Quotients

I think this is better -- if / when I write a page on universal properties, uniqueness will be automatic. I also thought it was confusing to have every extension of $F$ also being a quotient field of $D$. --Linus44 21:11, 23 February 2011 (CST)


 * The way I've been constructing pages is to make the definition as bald and simple as possible, i.e. "a" quotient field, then linking with a "see also" to different pages that provide the proofs that they exist and are unique.


 * We could change to include the existence/uniqueness proof on the same page as the definition, but I'm not sure this is always intrinsically a better approach.


 * Particularly in this case, where the construction of the quotient field is part and parcel of the fairly complicated inverse completion construction which applies in more than one context.


 * I'm fairly sure we've already got separate pages proving the existence / uniqueness result here anyway.


 * But whatever. If others have a different vision as to how this website is to develop, go for it. Just that every time we have a change of direction, we have to go back and change what's already there, and this is slowing down the development of new pages.


 * Just my 2-cents worth. --prime mover 00:20, 24 February 2011 (CST)


 * This is true. Its easy enough to change pages retrospectively that it makes sense to do so later if it is necessary, rather than for each user to begin from the beginning.


 * On the other hand I don't think that having terse definitions is always good, this often doesn't indicate exactly what the definition means.


 * I'll try not to get bogged down with unnecessary changed to what's already written, but I think that there is no harm in appending a discussion to existing pages. -- Linus44 08:03, 24 February 2011 (CST)


 * Yer all right. I looked at this again and yeh, nothing wrong with it. IMO pages are better short than long, as it's so easy to feel that everything and the kitchen sink needs to go in. But now I've taken time to have a good look at this, then yeah, no worries. --prime mover 15:25, 24 February 2011 (CST)


 * I've been thinking about this, I agree that relatively short pages are probably better. Sometimes there are significantly different ways of expressing a definition, in which case one can be a theorem, otherwise both probably aren't necessary.


 * I think anyway -- judging by the page histories you seem to have written a large proportion of the site yourself, so I'll defer to you on such matters. --Linus44 21:18, 26 February 2011 (CST)

Rename
No to the rename. We already have an "also known as" section, and there are a number of pages using that terminology. It would mean renaming them all, and I don't think it's worth the effort. --prime mover (talk) 08:19, 4 December 2016 (EST)

Definition is wrong
Is this how it is literally defined in those sources? $\Q(X)$ contains an isomorphic copy of $\Q(X)$, and so does the subfield $\Q(X^2)$, but $\Q(X^2)\neq \Q(X)$. --barto (talk) 07:33, 3 May 2017 (EDT)


 * No it's not. Warner's definition is how it was in 2008. In 2011 Linus44 changed it along with a lot of other stuff. I didn't question it because he seemed to know what he was doing.


 * Not that I understand what you'r saying either, dunno what you mean by $\Q(X^2)$. --prime mover (talk) 07:42, 3 May 2017 (EDT)


 * Ok. Either way Warner's definition has to be put back because we want all possible definitions. I see how to cure Linus44's one, and I'll add another one myself (using its universal property). --barto (talk) 07:51, 3 May 2017 (EDT)


 * $\Q(X^2)$ is the subfield of $\Q(X)$ generated by $X^2$. --barto (talk) 07:54, 3 May 2017 (EDT)


 * Any preferred order? --barto (talk) 07:55, 3 May 2017 (EDT)


 * Don't really care about the order. Warner's definition can be found in the History. The best one to take is the last change before Linus44's 2011 amendments. --prime mover (talk) 08:41, 3 May 2017 (EDT)


 * I put the simplest definitions first. I made a slight amendement to Warner's definition, by allowing for an embedding rather than an inclusion. I hope this is no problem, but I assume something has to be said about the difference. --barto (talk) 09:02, 3 May 2017 (EDT)

Definition as smallest field
As you may see, the current definitions do not use the terminology 'smallest'. Maybe there's a way to do so, by defining it as the smallest set in the class of fields containing $D$ or something, but I don't know much about that level of formalization in set theory. --barto (talk) 08:32, 3 May 2017 (EDT)

Notation
Please keep the notation, whether you like it or not: $\left({F, \oplus, \cdot}\right)$, $\left({D, +, \circ}\right)$. There is good reason for it to be there. PLease don't just change stuff because you don't like it. That is what I meant when I talked about refactoring. --prime mover (talk) 09:00, 3 May 2017 (EDT)


 * What I don't know is where you want those notations and where you don't want them. Only in the very first sentence or really everywhere? --barto (talk) 09:06, 3 May 2017 (EDT)


 * It emphasises the fact that the operations on the integral domain and the field are in fact different operations, as they are on different sets. I did not see fit to note this on the first version, where it was all $+, .$, but when Linus44 entered the picture, different notations were used to distinguish between the two.


 * I understand you want to make that distinction. But on the entire page, no operations are written, so I see no reason to specify them in this case. --barto (talk) 10:03, 3 May 2017 (EDT)


 * Despite that, I see no reason not to. --prime mover (talk) 10:05, 3 May 2017 (EDT)


 * I think it's less beautiful and harder to read. --barto (talk) 10:07, 3 May 2017 (EDT)


 * I disagree -- but it's not just a matter of personal preferences, it's part of our consistency of style. And besides, $\left({D, +, \circ}\right)$ is used on this page, down at the bottom. --prime mover (talk) 10:11, 3 May 2017 (EDT)


 * When a page appears standalone like this, without the user necessarily having previously gone through the full definitions of field and integral domain, it makes sense to include the "full specification" of the objects in question -- at least when first mentioned -- so as to place the user in the correct context.


 * Finally consistency. We use $D$ rather than $R$ to specify an integral domain. We just do. It's good to keep consistency. Also, we use $D^*$ (or is it just $D*$ to denote $D \setminus \{0\}$ as opposed to $D^\times$. Again, we just do.  (Although for numbers we have tended to use $D_{\ne 0}$ -- may also be appropriate here eventually). IMO -- IMVHO -- there is no need to just change notation if the existing notation works. --prime mover (talk) 09:22, 3 May 2017 (EDT)


 * Incidentally, can we change back to Monomorphism from Embedding? Everywhere else on the site Monomorohism is used, there is no need to use Embedding, at least in this context, because it compromises consistency. --prime mover (talk) 09:26, 3 May 2017 (EDT)

Definite article
The remark I made under also defined as was to address the use of a definite article (the quotient field) while we have not defined a privileged choice of the field. --barto (talk) 10:11, 3 May 2017 (EDT)


 * When (as was initially specified) the quotient field was defined, it was as a superset of the integral domain. In that context, the quotient field is unique. There can be only one. When you are just talking an arbitrary field into which you are embedding the I.D there are of course as many Q.Fs as there are these arbitrary fields, loosely speaking. So I still don't understand your point. --prime mover (talk) 10:14, 3 May 2017 (EDT)


 * Yes, uniqueness is easily misunderstood. The quotient field is unique in an algebraic or category-theoretic sense, not in a set-theoretic sense. We may artificially replace $\frac13$ by a new symbol in $\Q$ and still have a quotient field of $\Z$. When doing algebra, all we care about is its existence and its universal property; compare with tensor product. --barto (talk) 10:25, 3 May 2017 (EDT)

Rename
I've always used "Field of fractions", but knowing that personal preference is a bad argument, I did some searches to see what's most used, found that "quotient field" is. Went to see if this had been discussed at Wikipedia, and indeed: wikipedia:Talk:Field of fractions. Originally the page was called "Quotient field". Summary of their observations: (see also ciphergoth's concluding comment) In summary, I suggest to proceed as they did, by renaming to "Field of Fractions". --barto (talk) (contribs) 08:13, 4 November 2017 (EDT)
 * 1) "quotient field" is sometimes used for the quotient ring when it is a field
 * 2) It suggests that it is related to "quotient ring", "quotient group", ... but it is not
 * 3) Consistency with the analogue for rings that are not integral domains: "(total) ring of fractions" (and not, of course, "quotient ring")
 * 4) "quotient field" is used slightly more often than "field of fractions". This may be biased, because it does not take into account possible other uses of the term (see 1)


 * The problem is that a "fraction" is a colloquial term for a rational number which is not an integer, and there is a danger that the interpretation of "field of fractions" would suggest that the underlying ring was the integers.


 * If you really have reasons to object to "quotient field" (I don't share those reasons) then "field of quotients" would be a compromise which would work for me. I hold no respect for majority rule. --prime mover (talk) 08:28, 4 November 2017 (EDT)


 * Considering all the above, "field of quotients" has my preference. &mdash; Lord_Farin (talk) 14:57, 4 November 2017 (EDT)


 * As agreed, finally changed to Definition:Field of Quotients. Now the tidying up to be done.