if $a+bi$ is prime, $a- bi$ is also prime (Gauss integers) (irreducible)

Question

if a complex number is prime in Gauss integers, does it follow that its complex conjugate is also prime?

I know in general if a “regular” number divides $a+bi$, it also divided $a-bi$ but can’t show the same for all cause integers. Irreducibility is the same as being prime in the ring.

Gauss integers are the form $x+iy$ where $x$ and $y$ are integers.

Answer 1

Yes. Complex conjugation defines an automorphism of the ring $\mathbb Z[i]$. The image of a prime element by a ring automorphism, is again a prime element.

Answer 2

Yes, if $x$ and $y$ satisfy the conditions, then both $x+iy$ and $x-iy$ are Gaussian prime.

Please see Gaussian Prime. The conditions are on $x^2+y^2$, $|x|$, and $|y|$, so obviously $x+iy$ Gaussian prime $\iff x-iy$ Gaussian prime.