Show that Zn is a field if and only if n is a prime.

Please show steps. Thanks!

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- Apr 24th 2009, 11:17 AM #1

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- Apr 24th 2009, 12:55 PM #2
do you know what a field is? go through and prove that $\displaystyle \mathbb{Z}_n$ when $\displaystyle n$ is prime fulfills all the conditions. note that if n is not prime you will have zero divisors (you must show this), which of course, is bad, and would make the group not a field.

- Apr 24th 2009, 08:32 PM #3

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If $\displaystyle n=ab$ where $\displaystyle 0<a,b<n$ then $\displaystyle [a]_n[b]_n = [0]_n$ but $\displaystyle [a]_n,[b]_n\not = 0$ so it is not an integral domain.

If $\displaystyle n$ is prime then for any $\displaystyle x\not \equiv 0(\bmod n)$ there is $\displaystyle y$ with $\displaystyle xy\equiv 1(\bmod n)$ this means $\displaystyle [y]_n$ is inverse of $\displaystyle [x]_n$.