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Rational numbers, rational functions, and Gaussian rationals are examples of fields of fractions. In each of those cases, one knows what the quotients are long before one hears of the idea of constructing the field of fractions of an integral domain.

One case where one (typically??) does not know of such a thing in advance is the field of "convolution quotients"---the field of fractions of a ring of functions of a real variable in which the "multiplication" is convolution.

But convolution quotients will not be appreciated by students who just finished a first-semester calculus course last week. Is there some example one could mention to such students where they wouldn't think they already know what is meant by division of the objects in question?

Later edit suggested by answers and comments posted so far: I had in mind two or three purposes. One was that I wanted to mention this topic a bit obliquely in something the students are to read, and that had to be really terse, so I can't do anything really involved. Less than an hour after I posted the question, this ended up being a parenthetical comment on the course web site that said: "(for example, why is it that one can `divide' one divergent series by another?)". Here I had in mind the ring of formal power series suggested by Chris Eagle, but of course I needed to ruthlessly avoid mentioning power series.

A second purpose concerned possible future uses. Not only in courses: if we get some good examples here, I'd like to add them to Wikipedia's article titled "field of fractions".

A possible third purpose was just the satisfaction of knowing more than one decent example (since the only one mentioned above that's "decent" in the relevant sense is convolution quotients).

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Maybe formal power series and formal Laurent series? –  Chris Eagle Apr 26 '12 at 18:12
    
COnstruct the coordinate ring of some curve, define the ring of germs at a point, and then show that you can get it by localizing. –  Mariano Suárez-Alvarez Apr 26 '12 at 18:31
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@MarianoSuárez-Alvarez : I think your example suffers from the same difficulty that afflicts convolution quotients. –  Michael Hardy Apr 26 '12 at 18:38
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At some point thy will have to come to terms with the fact that the notions they are being exposed to are interesting for reasons that thy have not previously been exposed to. –  Mariano Suárez-Alvarez Apr 26 '12 at 19:05
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@Mariano: I remember one of my undergraduates teachers said once: "You know you are getting closer to finishing your degree when, instead of hearing a lot of "And this is important for reasons you will see in ..." you hear a lot of "And this is important, for reasons you saw in ...." –  Arturo Magidin Apr 27 '12 at 17:12

2 Answers 2

Ask them how many intermediate rings $\mathbb Z\subset R \subset \mathbb Q$ there are and whether they can classify them all.
Amaze them by telling them that there is a continuum ($=2^{\aleph_0}$) of them and that you can classify all of them explicitly: they are indexed by the subsets $P\subset \lbrace 2,3,5,7,\ldots \rbrace $ of the primes and they are the $S_P^{-1}\mathbb Z=\mathbb Z[\frac {1}{p}\mid p\in P]$ (where $S_P$ is the multiplicative monoid generated by $P$)

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Well, this answer obviously won't serve the immediate purpose. Students of the kind I have in mind are from this Reality, not some other. (I.e. computer science students whose professional interest is in things other than math. And they really haven't had anything beyond first-semester calculus before this. Really. I mean: really.) It's an interesting answer, but for the immediate purpose it can't be taken literally. –  Michael Hardy Apr 26 '12 at 19:13
    
....However, some version of this answer may well serve the less immediate purposes of the question. –  Michael Hardy Apr 26 '12 at 19:20
    
@Michael But these examples are neither fields nor examples of elements that students don't already know how to divide. Could you please clarify if either of these points are essential (as they seem to be in you query). –  Bill Dubuque Apr 26 '12 at 19:30
    
@BillDubuque : Well, definitely it's better to use examples of things where it's not obvious that division is possible. –  Michael Hardy Apr 26 '12 at 23:32
    
@Michael Note that formal rational "functions" are not really quotients of functions, so this example may already meet your requirements, i.e. they really are "formal" quotients. However, your students might not yet be ready to appreciate the distinction between formal vs. functional polynomials (and their fractions). –  Bill Dubuque Apr 26 '12 at 23:53

Here's a mean trick you can pull if they really did just finish first semester calculus. Introduce the ring $$ \mathbb{R}\left[\frac{d}{dx}\right] $$ which is of course just a polynomial ring, but don't say this. Show that it "acts'' on the space of smooth functions in the manner suggested by the notation.

Then ask a student what the field of fractions should be, suggestively saying "and what will the inverse of $\frac{d}{dx}$ be?"

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Well...... There's such a thing as $\dfrac{D}{e^D-1}$ (where of course $D=d/dx$, and it doesn't even involve a "constant of integration!, and in fact $\dfrac{D}{e^D - 1} x^n = B_n(x)$ $=\text{the $n$th-degree Bernoulli polynomial}$. But I still have some qualms about using this. –  Michael Hardy Apr 26 '12 at 18:53
    
Alternatively you could have the polynomial ring act on smooth functions mod constants, in which case the action really does extend to the field of fractions. –  user29743 Apr 26 '12 at 19:13

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