Results 1 to 3 of 3

Thread: Rings3

  1. #1
    Junior Member
    Joined
    Apr 2009
    Posts
    36

    Rings3

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

    Please show steps. Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    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 $\displaystyle \mathbb{Z}_n$ 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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    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 $\displaystyle n=ab$ 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 $\displaystyle [x]_n$.
    Follow Math Help Forum on Facebook and Google+

Search Tags


/mathhelpforum @mathhelpforum