I would appreciate help on completing the following exercise in my workbook:
If $K$ is a number field, $K/Q$ is a normal (Galois) extension, $O_K$ its ring of algebraic integers, $p$ is a prime number, and $pO_K$ represents the product of distinct prime ideals of norm $p$, then all of these ideals are principal, or they are all non-principal. If they are non-principal, then there are no elements of norm $p$. In this case, let $P$ be a prime ideal of norm $p$, and let $c$ be its class in the class group. If there is a fractional ideal $V$ of norm $1$ whose class is $c^{-1}$, then $VP$ will be a principal ideal of norm $p$, but its principal generators will be non-integer fractions in $K$.
If $R$ is a prime ideal in the "numerator" of $V$, there has to be a corresponding $R'$ conjugate to $R$ in the "denominator" (assuming $K/Q$ is a normal extension).
Any fractional ideal of norm $1$ is therefore a product of factors of the form $R/R'$ where $R$ is a prime ideal in $O\zeta_k$.
Exercise: Construct a principal ideal of norm $2$ in the field $K = \mathbb{Q}(√(-23))$ using the method described above.
My work:
The ideal $P = <2,√(-23)+1>$ is non-principal, therefore there are no elements of norm $2$. $P$ is already a prime ideal of norm $2$.
I am stuck at determining its class in the class group. Any help to finishing this please? Thanks.