Hw12 Problem 14

Return to Homework 12, Glossary, Theorems

Problem: (*) Show that the characteristic of an integral domain must be either $0$ or prime.

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Solution: Using Theorem 19.15 we can look solely at $mn * 1 = 0$ for $m > 1$ and $n > 1$, where $mn$ is the characteristic of the integral domain. Now,

(1)

\begin{equation} mn * 1 = (m * 1)(n * 1) \end{equation}

(2)

\begin{equation} = 0 \end{equation}

Since we are operating within an integral domain, either $m * 1 = 0$ or $n * 1 = 0$. So then the characteristic is both at most $m$ and at most $n$. Then the characteristic cannot be a composite integer. Therefore the characteristic of an integral domain is either $0$ or prime.