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Math Help - Rings3

  1. #1
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    Rings3

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

    Please show steps. Thanks!
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by mpryal View Post
    Show that Zn is a field if and only if n is a prime.

    Please show steps. Thanks!
    do you know what a field is? go through and prove that \mathbb{Z}_n when 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.
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  3. #3
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    Quote Originally Posted by mpryal View Post
    Show that Zn is a field if and only if n is a prime.

    Please show steps. Thanks!
    If n=ab where 0<a,b<n then [a]_n[b]_n = [0]_n but [a]_n,[b]_n\not = 0 so it is not an integral domain.

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