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If n is composite, then Z/(n) is not a field


Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.4

Show that if $n$ is not prime, then $\mathbb{Z}/(n)$ is not a field.


Solution: If $n$ is not prime, then some $1 \neq a < n$ divides $n$, hence $\mathsf{gcd}(a,n) \neq 1$.

By a previous exercise, then, there does not exist an element c such that $ac = 1 \pmod n$. So a does not have a multiplicative inverse; hence $\mathbb{Z}/(n)$ is not a field.


Linearity

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