I have 2 questions regarding quadratic integers.
1a. Assume that = exists in Q[Sqrt(3)]. How can we show that if = that = ? Also, how can we show that if = ), then = ? Lastly, if = , how can we show that = ?
I know that is prime and that there are three congruence classes (mod ): one with -1, 0, and 1.
1b. How can it be proven that if + = and x,y,z are quadratic integers in Q[Sqrt(-3)], then (as described in 1a) must divide of one x, y, or z?
My first guess at this one is to reduce the equation to modulo .
All help is appreciated!