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).
homotopy theory, (∞,1)-category theory, homotopy type theory
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Introductions
Definitions
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Basic facts
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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.
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.
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 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 $0$ or invertible, and it also implies that $0\neq 1$.
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 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 $x # y$ iff $x - y$ is invertible, then we obtain the notion of Heyting field. (As shown here, the ring operations become strongly extensional functions.) In addition to $0\# 1$, the condition then means that every element apart from $0$ 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 $0\neq 1$, this condition means that every noninvertible element (i.e. element $x$ such that $x y\neq 1$ for all $y$) 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. The following proof is due to Mark Saving:
Let $p$ be a proposition. We define the set $R_p$ to be the union of $\mathbb{Z}$ and $\{x \in \mathbb{Q} \vert p\}$
$R_p$ is a subring of $\mathbb{Q}$.
Since $R_p$ is a subset of $\mathbb{Q}$ and $\mathbb{Q}$ has decidable equality, $R_p$ also has decidable equality. And of course $0\neq 1$ in $R_p$.
$R_p$ is a residue field iff $\neg \neg p$.
Given a proposition $p$, suppose that $\neg \neg p$, and consider some $x\in R_p$. Suppose $x$ does not have a multiplicative inverse. Now suppose $x\neq 0$. Then we see that $x^{-1}$ is not in $R_p$. If $p$ held, we would have $x{-1} \in R_p$. So we know $\neg p$ holds. But this is a contradiction. Therefore, $x$ must be zero (using decidable equality).
Conversely, suppose $R_p$ is a residue field. If $\neg p$ held, we would have $R_p = \mathbb{Z}$, which clearly is not a residue field since $2$ is neither invertible nor zero. So we must have $\neg \neg p$.
$R_p$ is a Heyting field iff it is the case that $p$ iff $R_p$ is a discrete field.
Suppose $R_p$ is a Heyting field. Then either $2$ or $3$ has a multiplicative inverse, so either $2^{-1} \in R_p$ or $3^{-1} \in R_p$. In either case, we see that $p$ holds. If $p$ holds, then $R_p = \mathbb{Q}$, which is a discrete field. And if $R_p$ is a discrete field, it is clearly a Heyting field.
If every residue field with discrete equality is Heyting, then excluded middle is valid
From the lemmas above, if every residue field with decidable equality is a Heyting field, then $p \iff \neg \neg p$ holds for all propositions $p$. So we have full excluded middle.
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 $(0)$ is a prime ideal.
In LombardiQuitté2010, the authors’ definitions of discrete field and Heyting field do not include the non-equational axiom $1 \neq 0$ or $1 \# 0$ 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.
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.
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:
(all) algebraic numbers, (Kontsevich-Zagier) periods
p-adic numbers (for $p$ a prime number).
Other examples
finite fields $\mathbf{F}_{p^n}$ where $p$ is a prime
algebraic number fields – finite degree extensions of the field of rational numbers
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 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)
Ray Mines, Fred Richman, Wim Ruitenburg, A course in constructive algebra, Universitext, Springer, 1987.
Peter Johnstone, Sketches of an Elephant, Part D. The classifying topos for fields is discussed in section D3.1.11(b).
Peter Johnstone, Rings, Fields, and Spectra, Journal of Algebra 49 (1977) 238-260 doi
Olivia Caramello, Peter Johnstone, De Morgan’s law and the theory of fields (arXiv:0808.1972)
Henri Lombardi, Claude Quitté (2010): Commutative algebra: Constructive methods (Finite projective modules) Translated by Tania K. Roblo, Springer (2015) (doi:10.1007/978-94-017-9944-7, pdf)
Discussion in univalent foundations of mathematics (homotopy type theory):
Last revised on July 6, 2022 at 15:05:23. See the history of this page for a list of all contributions to it.