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\section*{field}

\begin{quote}%
This entry is about the notion in [[algebra]]. For the different concept of the same name in [[differential geometry]] see at \emph{[[vector field]]} and in [[physics]] see at \emph{[[field (physics)]]}.


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{linear_algebra}{}\paragraph*{{Linear algebra}}\label{linear_algebra}

[[!include homotopy - contents]]

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{fields}{}\section*{{Fields}}\label{fields}

\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{constructive}{Constructive notions}\dotfill \pageref*{constructive} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{category}{Category of fields}\dotfill \pageref*{category} \linebreak
\noindent\hyperlink{AccSketch}{Accessibility and sketchability}\dotfill \pageref*{AccSketch} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

Classically:

\begin{defn}
\label{classical}\hypertarget{classical}{}
A \textbf{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).

\end{defn}
Fields are studied in \emph{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 (algebra)|free field]] of Cohn and Amitsur is in fact noncommutative.

\hypertarget{constructive}{}\subsubsection*{{Constructive notions}}\label{constructive}

Fields are (arguably) not a purely [[algebra|algebraic]] notion in that they don't form an [[algebraic category]] (see \hyperlink{category}{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.

\begin{defn}
\label{discrete}\hypertarget{discrete}{}
If we replace ``an element is invertible iff it is nonzero'' in Definition \ref{classical} 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 \textbf{discrete field}. In addition to $0\neq 1$, this condition means that every element is either $0$ or invertible.

\end{defn}
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 logic|coherent theory]] and hence also a [[geometric theory]]; for this reason \hyperlink{Johnstone77}{Johnstone} calls such fields \textbf{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 number|rational]] (or even [[algebraic number|algebraic]]) numbers and by the [[finite field]]s as usual.

\begin{defn}
\label{heyting}\hypertarget{heyting}{}
If we interpret `nonzero' in Definition \ref{classical} 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 \textbf{Heyting field}. (As shown \href{/nlab/show/local+ring#internal}{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.

\end{defn}
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 number]]s need not form a Heyting field; another disadvantage is that this is not a coherent axiom and so cannot be [[internalization|internalized]] in as many categories.

\begin{defn}
\label{}\hypertarget{}{}
If we replace ``an element is invertible iff it is nonzero'' in Definition \ref{classical} 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 \textbf{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.

\end{defn}
An advantage is that even more versions of the [[real numbers]] (including the [[MacNeille real number]]s) 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 [[decidable equality|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]]. 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 \emph{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 number]]s form a residue field which need not be Heyting; see section D4.7 of \emph{[[Elephant|Sketches of an Elephant]]}.

The three definitions above do not exhaust the possible constructive notions of field. For instance, in \hyperlink{MRR87}{MRR87} the unadorned word \textbf{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 \hyperlink{Johnstone77}{Johnstone77} is called a \textbf{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 \hyperlink{MRR87}{MRR87} a \textbf{denial field} is defined to be a field of fractions in which additionally $(0)$ is a [[prime ideal]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{category}{}\subsubsection*{{Category of fields}}\label{category}

Fields are not as well-behaved [[category theory|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 category|complete]] or [[cocomplete category|cocomplete]], although it is [[accessible category|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 field]]s). In particular, it is therefore not [[algebraic category|algebraic]] or [[locally presentable category|locally presentable]].

\hypertarget{AccSketch}{}\subsubsection*{{Accessibility and sketchability}}\label{AccSketch}

The [[category]] of fields is [[accessible category|accessible]], even \emph{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

\begin{displaymath}
F \times F \;\rightrightarrows\; F,
\end{displaymath}
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

\begin{displaymath}
\itexarray{P & \overset{}{\to} & * \\
  \downarrow && \downarrow^0\\
  I& \hookrightarrow & F,}
\end{displaymath}
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 \emph{discrete} field (see the \hyperlink{constructive}{constructive definitions} above). The other constructive notions of field can also be described as models for different limit-colimit sketches.

\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\begin{example}
\label{}\hypertarget{}{}
There are the fields of:

\begin{itemize}%
\item [[rational numbers]],


\item [[algebraic numbers]],


\item [[real numbers]],


\item [[complex numbers]],


\item [[p-adic numbers]] (for $p$ a [[prime number]]).


\item Also the various [[finite fields]].



\end{itemize}
\end{example}
\begin{example}
\label{}\hypertarget{}{}
The canonical [[ring object|local ring object]] of the [[Zariski site|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 \ref{classical} holds in the internal logic of this topos. For a proof, see Proposition 2.2 in the article \href{http://www.sciencedirect.com/science/article/pii/0022404976900025}{Universal projective geometry via topos theory} of Anders Kock. The ring $\mathcal{O}_X$ (the [[structure sheaf]]) in the [[category of sheaves|sheaf topos]] (i.e. the petit Zariski topos) is a [[residue field]] if $X$ is a [[reduced scheme|reduced]] scheme.

\end{example}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[division ring]]


\item [[subfield]]


\item [[ring]]


\item [[finite field]],


\item [[residue field]]


\item [[global field]]

\begin{itemize}%
\item [[number field]]


\item [[function field]] (over a finite field)



\end{itemize}

\item [[positive characteristic]]


\item [[topological field]]


\item [[field with one element]]


\item [[graded field]]


\item [[infinity-field]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Ray Mines]], [[Fred Richman]], [[Wim Ruitenburg]]. \emph{A Course in Constructive Algebra}. Springer, 1987.


\item [[Peter Johnstone]], [[Sketches of an Elephant]], Part D. The [[classifying topos]] for fields is discussed in section D3.1.11(b).


\item [[Peter Johnstone]], \emph{Rings, Fields, and Spectra}, Journal of Algebra \textbf{49} (1977) pp 238-260. doi:\href{https://doi.org/10.1016/0021-8693%2877%2990284-8}{10.1016/0021-8693(77)90284-8}


\item [[Olivia Caramello]], [[Peter Johnstone]], \emph{De Morgan's law and the theory of fields} (\href{http://arxiv.org/abs/0808.1972}{arXiv:0808.1972})



\end{itemize}
[[!redirects field]] [[!redirects fields]]

[[!redirects discrete field]] [[!redirects discrete fields]]

[[!redirects Heyting field]] [[!redirects Heyting fields]] [[!redirects heyting field]] [[!redirects heyting fields]]



\end{document}