This entry is about the notion in algebra. For the different concept of the same name in differential geometry see at vector field and in physics see at field (physics).
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
symmetric monoidal (∞,1)-category of spectra
Classically:
A field is a commutative ring in which every nonzero element has a multiplicative inverse and $0 \neq 1$ (which may be combined as: an element is invertible if and only if it is nonzero).
Fields are studied in field theory which is a branch of commutative algebra.
Not assuming the commutativity axiom, then the result is a skewfield or division ring (also in some contexts simply called a “field”). For example, the free field of Cohn and Amitsur is in fact noncommutative.
Fields are (arguably) not a purely algebraic notion in that they don't form an algebraic category (see discussion below). For this reason, it should be unsurprising that in constructive mathematics (including the internal logic of a topos) there are different inequivalent ways to define a field. In this case the classical 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” in Definition 1 by “an element is invertible xor it equals zero” (which is equivalent in classical logic but stronger in constructive logic), then we obtain the notion of discrete field.
Such a field $F$ is ‘discrete’ in that it decomposes as a coproduct $F = \{0\} \sqcup F^\times$ (where $F^\times$ 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 these are defined), although it is satisfied by the ring of rational (or even algebraic) numbers and by the finite fields as usual.
If we interpret ‘nonzero’ in Definition 1 as a reference to a tight apartness relation, thus defining the apartness relation $\#$ by $x # y$ iff $x - y$ is invertible, then we obtain the notion of Heyting field. (As shown here, the ring operations become strongly extensional functions.)
This is how ‘practising’ constructive analysts of the Bishop school usually define the simple word ‘field’. An advantage is that the (located Dedekind) real numbers form a Heyting field, although (for example) the (less located) MacNeille real numbers 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 we replace “an element is invertible iff it is nonzero” in Definition 1 by “an element is noninvertible iff it is zero” (which is equivalent in classical logic but incomparable in constructive logic), we obtain the notion of residue field (which is not quite the same as the residue fields of algebraic geometry).
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 (located 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.
Fields are not as well-behaved categorically as most other common algebraic structures (groups, rings, modules, etc.). In particular, the category of fields and field homomorphisms (a full subcategory of the category of rings) is not complete or cocomplete, although it is accessible.
In particular, it lacks a terminal object and also lacks an initial object (though it has a weakly initial set, namely the set of prime fields). In particular, it is therefore not algebraic or locally presentable.
The category of fields is accessible, even finitely accessible, and therefore can be presented as the category of models (in Set) of a mixed limit-colimit sketch. It is moreover straightforward to write down such a sketch.
We suppose as given to start with a limit sketch? whose models are commutative rings, with $F$ denoting the ring. We can construct via limit constructions a subobject $I\hookrightarrow F$ consisting of the invertible elements, as the equalizer of the two maps
the first being given by multiplication and the second by the composite $F\times F \to * \overset{1}{\to} F$, where $*$ is terminal and the map labeled “1” picks out the element $1\in F$. We now assert that if we take the pullback
where the map labeled “0” picks out the element $0\in F$, then the object $P$ is initial (i.e. $0$ is not invertible, or equivalently not equal to $1$), and moreover the pullback is also a pushout (i.e. every element of $F$ is either $0$ or invertible). Of course, in making these last two assertions we use the fact that we are allowing ourselves a limit-colimit sketch instead of just a limit sketch.
Note that this gives us the notion of discrete field (see the constructive definitions above). The other constructive notions of field can also be described as models for different limit-colimit sketches.
There are the fields of:
p-adic numbers (for $p$ a prime number).
Also the various finite fields.
The canonical local ring object of the gros Zariski topos of any scheme (given by $S \mapsto \Gamma(S, \mathcal{O}_S)$, that is to say, the affine line $\mathbb{A}^{1}_{S}$) is in fact moreover a field object, where the latter is defined by requiring that Definition 1 holds in the internal logic of this topos. For a proof, see Proposition 2.2 in the article Universal projective geometry via topos theory of Anders Kock. The ring $\mathcal{O}_X$ (the structure sheaf) in the sheaf topos (i.e. the petit Zariski topos) is a residue field if $X$ is a reduced scheme.
function field (over a finite field)
The classifying topos for fields is discussed in section D3.1.11(b) of