Question

Definition:In the complex numbers, let J denote the set, {x+y√3i :x and y are in Z}. J is an integral domain containing Z. If a is in J, then N(a) is a non-negative member of Z. If a
and b are in J and a|b in J, then N(a)|N(b) in Z. The units of J are 1, -1

Question:If a and b are in J and ab = 2, then prove one of a and b is a unit. Thus, 2 is prime in J. Show −2 is also prime.

Answer 1

a1bEJ, ab = 2

Then $N(ab)=N(a)N(b)=N(2)=4$

As
4-1×4-2×2
if neither of a and b are units we must have $N(a)=N(b)=2$

If $a=x+iy\sqrt{3}\Rightarrow N(a)=x^2+3y^2\neq 2$ as
2
is not possible which makes $x^2=2$ which is also impossible

So meaning at one of them is a unit

or Л.lb

Similarly, if $a,b\in J,ab=-2\Rightarrow N(ab)=N(a)N(b)=N(-2)=4$ and similar logic tells us that

meaning at one of them is a unit

or Л.lb

Thus, 2 and -2 are primes in J