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Contents Embed Dark Mode Prev Up Next \(\newcommand{\N}{\mathbb N}
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Section Characteristic of a ring
In this section we define the
characteristic of a ring that contains a subfield .
Definition 14 .
Let
\(f\colon\mathbb{Z}\to R\) be a unique ring homomorphism. The kernel of
\(f\) is generated by the unique positive integer
\(n\text{,}\) \(\ker f=(n)\text{.}\) The integer
\(n\) is called the
characteristic of
\(R\text{.}\)
Checkpoint 15 .
Find the characteristic of the following fields.
\begin{equation*}
\Q,\R,\C,\F_p,\F_p(t),\Q(t)
\end{equation*}
Checkpoint 16 .
Find the characteristic of the following rings.
\begin{equation*}
\R[x]/(x^2+1),\quad\Q[x]/(x^3-2),\quad\F_3[x]/(x^2+1)
\end{equation*}
