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\begin{document}

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\section*{characteristic}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

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

[[!include higher algebra - contents]]

\hypertarget{the_characteristic_of_a_field_etc}{}\section*{{The characteristic of a field (etc)}}\label{the_characteristic_of_a_field_etc}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{for_rings}{For rings}\dotfill \pageref*{for_rings} \linebreak
\noindent\hyperlink{for_rings_2}{For $E_\infty$-rings}\dotfill \pageref*{for_rings_2} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \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{idea}{}\subsection*{{Idea}}\label{idea}

It is well known that you cannot divide by [[zero]], lest you be doomed to [[trivial ring|triviality]]. Conversely, in a [[field]], you \emph{can} divide by anything \emph{except} zero. But this rule can be misleading, since it's possible that (even) an ordinary number can be zero when you don't expect it! The \emph{characteristic} of a field states when (if ever) this happens.

It is straightforward to generalise from fields to other [[rings]], and even [[rigs]]. See also [[characteristic zero]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{for_rings}{}\subsubsection*{{For rings}}\label{for_rings}

Let $K$ be a [[rig]] (possibly a [[ring]], possibly a [[commutative ring]], possibly even a [[field]]). Then there exists a unique [[homomorphism]] $\phi_K\colon \mathbb{N} \to K$ to $K$ from the [[initial object|initial]] rig, which is the rig $\mathbb{N}$ of [[natural numbers]]. The [[kernel]] of $\phi_K$ is an [[ideal]] of $\mathbb{N}$, which (by a well-known property of $\mathbb{N}$) is a [[principal ideal]] with a unique generator. This generator is the \textbf{characteristic} of $K$, denoted $\char K$.

If $K$ is a ring, then we may use $\phi_K\colon \mathbb{Z} \to K$ instead, where $\mathbb{Z}$ is the ring of [[integers]]. However, in this case, the kernel will usually have two generators, in which case we pick the positive one to get the same result as above.

\hypertarget{for_rings_2}{}\subsubsection*{{For $E_\infty$-rings}}\label{for_rings_2}

The concept of the characteristic has been generalized to [[E-∞ rings]] (\hyperlink{Szymik12}{Szymik 12}, \hyperlink{Szymik13}{Szymik 13}, \hyperlink{Baker14}{Baker 14}).

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

If $n$ is a [[natural number]], then we suppress mention of $\phi_K$ to think of $n$ as an element of $K$. If $K$ is a ring, then we do the same for a negative [[integer]] $n$. We then have that $n = 0$ in $K$ if and only if $n$ is a multiple of $\char K$.

The characteristic of a field must be either [[zero]] or a [[prime number]]. Basically, this is because the kernel of $\phi_K$, for $K$ a field, must be a [[prime ideal]].

Every rig with positive characteristic is in fact a ring, since we have $\char K - 1 = -1$. In other words, any rig other than a ring must have characteristic zero (although many rings also have that characteristic).

If there is any [[homomorphism]] at all between two fields, then they have the same characterstic. In other words, any [[field extension|extension]] of a field keeps the same characteristic.

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

If $n$ is a positive natural number, then the characteristic of $\mathbb{N}/n = \mathbb{Z}/n$ is $n$. This ring is always a [[commutative ring]], and it is a [[field]] if and only if $n$ is prime, in which case it is the [[prime field]] $\mathbb{F}_n$. More generally, every [[finite field]] has positive prime characteristic.

For $n = 0$, $\mathbb{N}/0 = \mathbb{N}$, $\mathbb{Z}/0 = \mathbb{Z}$, and the prime field $\mathbb{F}_0 = \mathbb{Q}$ (the field of [[rational numbers]]) are no longer all the same, but they still all have characteristic $0$. Every [[ordered field]] has characteristic $0$. The [[real numbers]] and [[complex numbers]] each form fields of characteristic $0$.

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

\begin{itemize}%
\item [[freshman's dream]]


\item [[characteristic zero]], [[positive characteristic]]



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

Discussion for [[E-∞ rings]]:

\begin{itemize}%
\item [[Markus Szymik]], \emph{Commutative S-algebras of prime characteristics and applications to unoriented bordism} (\href{http://arxiv.org/abs/1211.3239}{arXiv:1211.3239})


\item [[Markus Szymik]], \emph{String bordism and chromatic characteristics} (\href{http://arxiv.org/abs/1312.4658}{arXiv:1312.4658})


\item [[Andrew Baker]], \emph{Characteristics for $E_\infty$ ring spectra} (\href{http://arxiv.org/abs/1405.3695}{arXiv:1405.3695})



\end{itemize}
[[!redirects characteristic]] [[!redirects characteristics]] [[!redirects characteristic of a field]] [[!redirects characteristics of fields]] [[!redirects characteristic of a ring]] [[!redirects characteristics of rings]] [[!redirects characteristic of a rig]] [[!redirects characteristics of rigs]]



\end{document}