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\noindent {\bf Question}
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\noindent Let $n\ge 4$ be an integer that is not prime.  Show that
the integers modulo $n$, ${\bf Z}_n$, is not a field.

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\noindent {\bf Answer}

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\noindent Write $n$ as a product $n =a\cdot b$, where $2\le a, b
<n$, so that $a$ and $b$ are not equal in ${\bf Z}_n$.  Then, in
${\bf Z}_n$, the product $a\cdot b$ is $0$, being a multiple of
$n$. However, if ${\bf Z}_n$ were a field, then $a$ would have a
multiplicative inverse $a^{-1}$, and we could multiply both sides
of $a\cdot b =0$ on the left to obtain $a^{-1}\cdot a\cdot b =
a^{-1}\cdot 0$, which simplifies to $b = 0$.  This contradicts the
choice of $b$ to satisfy $2\le b < n$, and so $a$ has no
multiplicative inverse, contradicting the definition of a field.


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