nLab field


This entry is about the notion in algebra. For the different concept of the same name in differential geometry see at vector field, and for that in physics see at field (physics).


Linear algebra

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



Paths and cylinders

Homotopy groups

Basic facts







A field is a commutative ring in which every nonzero element has a multiplicative inverse and 010 \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.

If we omit 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.

Constructive notions

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. These include:


If we replace “an element is invertible iff it is nonzero” in Definition 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. This condition means that every element is either 00 or invertible, and it also implies that 010\neq 1.

Such a field FF is ‘discrete’ in that it decomposes as a coproduct F={0}F ×F = \{0\} \sqcup F^\times (where F ×F^\times is the subset of invertible elements). An advantage is that this is a coherent theory and hence also a geometric theory; for this reason Johnstone calls such fields geometric fields. A disadvantage is that this axiom 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 as a reference to a tight apartness relation, thus defining the apartness relation #\# by x#yx # y iff xyx - y is invertible, then we obtain the notion of Heyting field. (As shown here, the ring operations become strongly extensional functions.) In addition to 0#10\# 1, the condition then means that every element apart from 00 is invertible.

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 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). In addition to 010\neq 1, this condition means that every noninvertible element (i.e. element xx such that xy1x y\neq 1 for all yy) is zero.

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 field is a discrete field if and only if equality is decidable; it is in this sense that a discrete field is ‘discrete’. It is not true that every residue field with decidable equality is Heyting. See this proof for details.

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. 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. The name “residue field” comes from the fact that these fields are precisely the residue rings of weak local rings (rings in which the noninvertible elements form an ideal).

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.

The three definitions above do not exhaust the possible constructive notions of field. For instance, in MRR87 the unadorned word field is defined like a Heyting field above, but with #\# being an arbitrary inequality relation rather than a tight apartness. If the inequality is the denial inequality, this reproduces the original classical definition, which in Johnstone77 is called a field of fractions since they are precisely the fields of fractions of “weak integral domains” (defined as rings in which the product of two nonzero elements is nonzero). In MRR87 a denial field is defined to be a Heyting field with respect to the denial inequality in which additionally (0)(0) is a prime ideal.


In LombardiQuitté2010, the authors’ definitions of discrete field and Heyting field do not include the non-equational axiom 101 \neq 0 or 1#01 \# 0 respectively. With such a definition, the trivial ring counts as a discrete field as well as a Heyting field and constitutes the terminal object in the categories of such fields.


Category of fields

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 Rings of rings and ring homomorphisms) 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, hence has a “multi-initial object”). In particular, it is therefore not algebraic or locally presentable.

Accessibility and sketchability

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 FF denoting the ring. We can construct via limit constructions a subobject IFI\hookrightarrow F consisting of the invertible elements, as the equalizer of the two maps

F×FF, F \times F \;\rightrightarrows\; F,

the first being given by multiplication and the second by the composite F×F*1FF\times F \to * \overset{1}{\to} F, where ** is terminal and the map labeled “1” picks out the element 1F1\in F. We now assert that if we take the pullback

P * 0 I F,\array{P & \overset{}{\to} & * \\ \downarrow && \downarrow^0\\ I& \hookrightarrow & F,}

where the map labeled “0” picks out the element 0F0\in F, then the object PP is initial (i.e. 00 is not invertible, or equivalently not equal to 11), and moreover the pullback is also a pushout (i.e. every element of FF is either 00 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:

Other examples


The canonical local ring object of the gros Zariski topos of any scheme (given by SΓ(S,𝒪 S)S \mapsto \Gamma(S, \mathcal{O}_S), that is to say, the affine line 𝔸 S 1\mathbb{A}^{1}_{S}) is in fact moreover a field object, where the latter is defined by requiring that Definition 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 𝒪 X\mathcal{O}_X (the structure sheaf) in the sheaf topos (i.e. the petit Zariski topos) is a residue field if XX is a reduced scheme.

commutative ringreduced ringintegral domain
local ringreduced local ringlocal integral domain
Artinian ringsemisimple ringfield
Weil ringfieldfield


Discussion in univalent foundations of mathematics (homotopy type theory):

Last revised on January 12, 2023 at 17:06:58. See the history of this page for a list of all contributions to it.