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Math Help - Number Theory and Systems of Numbers!

  1. #1
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    Number Theory and Systems of Numbers!

    HeY ppl! Can any1 help me wit eitha of these 2 questions plzzzzzzzzzz?!?!?! I'm in abit of a pickle! I'm rely rely lost! The question itself is confusing!! ><

    1) Prove that a non-zero element in [Z]m is a zero divisor if and only if it is not relatively prime to m.

    2) Let Zp[i] denote the ring {a + bi | a,b E [Z]p, i^2=-1}.
    a) Show that if p is not prime, then Zp[i] is not an integral domain.
    b) Suppose p is prime. Show that every non-zero element of Zp[i] is a unit iff x^2 + y^2 does not equal 0modp for any pairs x,y E [Z]p.

    (Its a baby 'm' and 'p' next to the z's... couldn't find how to shrink letters! :S)

    Hope all u smart ppl can help me!!!!!!!!!! Urgently!!!!!!!!!!
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  2. #2
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    Quote Originally Posted by carmz View Post
    1) Prove that a non-zero element in [Z]m is a zero divisor if and only if it is not relatively prime to m.
    If [a]_m is not relatively prime to m then (m/d)d = m where d=\gcd(m,a)>1. Thus, [m/d]_m and [a]_m are non-zero elements and yet [m/d]_m \cdot [d]_m = [m]_m = [0]_m is a zero element.
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  3. #3
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    Thanks heaps for replying!!! ... Just a lil confused! How come you can conclude that and are non-zero elements? ...And with the last step, how come you make m=0 when in the question you need to show that a 'non-zero' element in [Z]m is a zero divisor. Sorry bout all the qs! I'm a lil thick n slow when it comes to proofs ><
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  4. #4
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    Quote Originally Posted by carmz View Post
    Thanks heaps for replying!!! ... Just a lil confused! How come you can conclude that and are non-zero elements? ...And with the last step, how come you make m=0 when in the question you need to show that a 'non-zero' element in [Z]m is a zero divisor. Sorry bout all the qs! I'm a lil thick n slow when it comes to proofs ><
    Remember what zero divisors mean. It means that ab=0 where a,b are non-zero. Since d>1 ot means [d]_m\not = [0] and [m/d]_m\not = [0] but their product is zero.
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  5. #5
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    I would love to know the answer to part 2 of this, if anyone knows please post.
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