In Z[i], show that 1+i divides a+bi precisely when a is congruent to b mod 2.

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- Apr 21st 2010, 02:29 PMNikoBellicIn Z[i], show that 1+i divides...
In Z[i], show that 1+i divides a+bi precisely when a is congruent to b mod 2.

- Apr 21st 2010, 02:37 PMchiph588@
$\displaystyle \frac{a+bi}{1+i} = \frac{a+b}{2} - \left(\frac a2 - \frac b2\right)i $ since $\displaystyle \frac{1}{1+i} = \frac{1-i}{2} $.

So we need $\displaystyle a+b $ and $\displaystyle a-b $ to be even. What does this tell us?