# Number Theory and Systems of Numbers!

• Apr 6th 2008, 08:27 PM
carmz
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!!!!!!!!!! :(:(:(
• Apr 7th 2008, 07:35 AM
ThePerfectHacker
Quote:

Originally Posted by carmz
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 $\displaystyle [a]_m$ is not relatively prime to $\displaystyle m$ then $\displaystyle (m/d)d = m$ where $\displaystyle d=\gcd(m,a)>1$. Thus, $\displaystyle [m/d]_m$ and $\displaystyle [a]_m$ are non-zero elements and yet $\displaystyle [m/d]_m \cdot [d]_m = [m]_m = [0]_m$ is a zero element.
• Apr 7th 2008, 05:48 PM
carmz
Thanks heaps for replying!!! :D:D:D... Just a lil confused! How come you can conclude that http://www.mathhelpforum.com/math-he...a519cbe6-1.gif and http://www.mathhelpforum.com/math-he...1cdde2c7-1.gif 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 ><
• Apr 7th 2008, 06:23 PM
ThePerfectHacker
Quote:

Originally Posted by carmz
Thanks heaps for replying!!! :D:D:D... Just a lil confused! How come you can conclude that http://www.mathhelpforum.com/math-he...a519cbe6-1.gif and http://www.mathhelpforum.com/math-he...1cdde2c7-1.gif 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 $\displaystyle ab=0$ where $\displaystyle a,b$ are non-zero. Since $\displaystyle d>1$ ot means $\displaystyle [d]_m\not = [0]$ and $\displaystyle [m/d]_m\not = [0]$ but their product is zero.
• Apr 7th 2008, 06:54 PM
craigoss
I would love to know the answer to part 2 of this, if anyone knows please post.