Can someone point me to resources that could either give examples of complex roots of unity which also happen to be complex primes (Eisenstein primes, Gauss primes, or any other type if they exist) or a proof that such complex numbers can’t exist?
Have searched google but couldn’t find information on such intersection of the two types of complex numbers.
Thanks!
On
You'd have to accept $1$ as a prime number. By the way, I'm so old I was taught as a child that $1$ is a prime number. It wasn't until very recently (a few years ago) that I learned better.
In order for an algebraic integer (whether it be complex or not) to be a prime in the relevant ring, it has to have a prime norm in that ring. So, for example, $1 + i$ has a norm of $2$, which is prime in $\mathbb{Z}$, so $1 + i$ is a Gaussian prime.
It should be a well-known fact that the norm is a multiplicative function. e.g., $N((1 + i)^2) = 4$ since $2 \times 2 = 4$. Now, $-i$ is a biquadrate complex root of unity, and it has norm $1$. Well, so does $(-i)^2$.
With complex algebraic integers, $-1$ is not a possible norm. A complex root of unity has norm $1$ in the relevant ring. Therefore it can't be a prime in that ring, because it's a unit.
If $1 \in A \subset \mathbb C$ is a subring with $\zeta \in A$, $\zeta^n=1$ in $\mathbb C$, then also $\zeta^n=1$ in $A$, so $\zeta$ is invertible in $A$.