Norm in the cyclotomic field

Question

Let $F = \mathbb{Q}(\xi_p)$ be the $p^{th}$ cyclotomic field.

What is the norm of $N(1 + \xi_p)$?

I’ve figured out that $N(1-\xi_p) = p$, as this can easily be seen from the minimal polynomial of $\xi_p$.

I’m stuck on how to find $N(1+\xi_p)$, though.

Answer 1

I presume $p$ is a prime. If $p\ne 2$ then $$N(1+\xi_p)=\frac{N(1-\xi_p^2)}{N(1-\xi_p)}=1$$ since $1-\xi_p^2$ is a conjugate of $1-\xi_p$.