In classical mathematics, a field is a commutative ring in which every nonzero element has a multiplicative inverse and (which may be combined as: an element is invertible if and only if it is nonzero). They are studied in field theory which is a branch of commutative algebra. If one drops commutativity, then the result is a skewfield, though sometimes said for simplicity just field. For example, the free field of Cohn and Amitsur is in fact noncommutative.
Fields are not as well-behaved categorically as most other common algebraic structures (groups, rings, modules, etc.). In particular, the category of fields is not complete or cocomplete, although it is accessible.
The words ‘field’ and ‘field theory’ are also used in mathematical physics and geometry with a completely different meaning denoting a quantity over some space, thus vector fields, classical scalar fields, classical tangent vector fields, quantum field?s and so on. Probably these should be summarised at something like physical field.
For the same reason, in constructive mathematics (such as the internal logic of a topos) there are different inequivalent ways to define a field. In this case the above definition is not usually the best one; for instance, the real numbers do not satisfy it. There are several potential replacements with their own advantages and disadvantages.
If we replace “an element is invertible iff it is nonzero” by “an element is invertible xor (exclusive or) it equals zero” (which is classically equivalent but constructively stronger), we obtain the notion of discrete field. Such a field is discrete in that it decomposes as a coproduct (where is the subset of invertible elements). An advantage is that this is a coherent theory; a disadvantage is that it is not satisfied (constructively) by the ring of real numbers (however defined), although it is satisfied by the ring of rational (or even algebraic) numbers and by the finite fields as usual.
If, alternatively, we interpret ‘nonzero’ in this clause as a reference to an apartness relation and assume that the ring operations are strongly extensional, then we obtain the notion of Heyting field. (Although this seems to have extra structure, the apartness relation is definable from the algebra: iff is invertible.) This is how ‘practising’ constructive analysts of the Bishop school usually define the simple word ‘field.’ An advantage is that the real numbers (in the located Dedekind sense) form a Heyting field, although (for example) the MacNeille real numbers (a less located version of the Dedekind reals) need not form a Heyting field; another disadvantage is that this is not a coherent axiom and so cannot be internalized in as many categories.
If, finally, we replace this clause by “an element is noninvertible iff it is zero” (which is classically equivalent but constructively incomparable), we obtain the notion of residue field. An advantage is that even more versions of the real numbers (including the MacNeille real numbers) form a residue field; disadvantages are that this axiom is not coherent either and that a residue field lacks an apartness relation (in particular, the MacNeille reals have no apartness).
Every discrete field is also a Heyting field, and every Heyting field is also a residue field. A Heyting or residue field is a discrete field if and only if equality is decidable; it is in this sense that a discrete field is ‘discrete’.
A residue field is a Heyting field if and only if it is a local ring. Furthermore, the quotient ring (or ‘residue ring’) of any local ring by its ideal of noninvertible elements is a Heyting field; in particular, it is a residue field (hence that name). On the other hand, not every residue field is even a local ring (the MacNeille reals are not), so not every residue field is the residue ring of any local ring.
Counterexamples were remarked above, but to be explicit: The (Dedekind) real numbers form a Heyting field which need not be discrete. The MacNeille real numbers form a residue field which need not be Heyting; see section D4.7 of Sketches of an Elephant.
This discussion was originally at vector space, but most of it is really about fields.
Eric Forgy says: Is there a nice arrow theoretic way to define vector spaces?
Urs says: There is comparatively nice abstract nonsense to be said about rings. For instance, a ring is a category with one object enriched in the category of abelian groups. From that starting point the concepts of ring theory develop rather naturally from pure category theoretic reasoning. In particular modules over rings appear naturally.
For some reason this is different when rings are refined to fields and modules to vector spaces. The very concept of a field is somehow not as natural from a category theoretic perspective, or at least I don’t see how it is. This problem becomes very manifest when one tries to categorify fields and vector spaces: it is very straightforward to categorify rings and their modules, but their refinement to categorified fields and vector spaces is harder.
Eric says: *tongue in cheek* Well, fields are closely tied to the continuum and we know the continuum is not natural :)
Toby says: *tongue where it belongs* A field may be defined a ring that has been decomposed (as a set) as the sum of a point and a group, with the point being the zero element and the the group being a multiplicative submonoid. However, this definition does not work in constructive analysis; one cannot prove constructively that the ring of real numbers has this property. Instead, one must define a field to be a ring equipped with an apartness relation satisfying some properties (which I can write down if desired). Classically, an apartness relation is a vacuous structure, but in constructive mathematics its properties have a very topological flavour, even though it's much simpler than a topology. All of the deep issues about the continuum are already there in the constructivists' notion of apartness, which they require to even define what a field is.
John says: As Toby notes, fields are a somewhat awkward concept in constructive mathematics. The basic problem is that their definition involves a clause “if , then has an inverse…” A clause of the for “if something is not true, then…” is rather annoying in constructive mathematics. This means fields are not very nice if you’re trying to do math in an arbitrary topos. And since topos theory was invented when Grothendieck was working on algebraic geometry, this should mean that fields are not very nice when you’re trying to do algebraic geometry!
Of course this is a bit shocking, since the first thing an elementary text on algebraic geometry does is say “Let be a field.” But Grothendieck’s revolutionary approach to algebraic geometry also downplays the importance of fields, so presumably it all hangs together once you understand it.
I wish I understood this better. There’s a nice exposition of a little bit of these ideas in Mac Lane and Moerdijk’s Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Look up ‘local ring’ in their index: you’ll see that unlike the notion of field, the notion of local ring can be defined using ‘geometric logic’, which is (very roughly) the kind of logic that works well in topoi.
Eric says: The wikipedia article on Category of fields has some interesting things to say about the difficulties with fields in category theory.
If fields are less than perfectly natural, then vector spaces are somewhat troublesome too. But what about the tangent bundle of a category? That idea is very pretty. In what way does it fail to converge to the usual concept of a tangent space, which should contain vector spaces (obviously)? [Edit: Sorry. The answer was in the very link I provided, but any further light on the subject would be appreciated.]
Toby says: Rather than think about the category of fields, shouldn't we think of fields as nice objects in the nice category of rings? Modules, after all, behave beautifully.
John says: Unlike the category of fields, the category of vector spaces is not troublesome at all! But there’s no contradiction here. When Eric wrote “fields are less than perfectly natural”, he was being a bit vague. Fields are wonderfully well-behaved rings, and rings form a wonderfully well-behaved category. But the category of fields is not so nice. As Toby notes, this is completely typical. So, if we fix a field and take the category of vector spaces over it, that category is wonderfully nice.
I have a question. This article says the category of fields is ‘accessible’, and it says that an accessible category is the category of models of a sketch. What sort of sketch does the job for fields. They’re not models of a finite limits theory, so I assume we also need some colimits in our sketch. How does this work? Which constructive definition(s) of field do we use?
Gonçalo Marques?: The sketch for fields is described on page 248 of “Category theory for computing science” by Michael Barr and Charles Wells (3rd edition). They use what they call “finite discrete sketches”. It does involve a cocone.