Given a prime $p$, the $p$-adic valuation on the field $\mathbb{Q}$ is the map $\nu:\mathbb{Q}^*\to\mathbb{Z}$ given by $\nu(p^ka/b)=k$, where $a,b$ are prime to $p$.
I want to consider extensions of this valuation to number fields; in particular I want to extend $\nu$ to a valuation on $\mathbb{Q}(\zeta_p)$, where $\zeta_p$ is a primitive $p$th root of unity.
Let $\mathcal{O}$ denote the ring $\mathbb{Z}[\zeta_p]$ of integers in $\mathbb{Q}(\zeta_p)$.
- Is it true that every prime ideal $\mathfrak{p}$ dividing $p\mathcal{O}$ determines an extension of the valuation?
- If so, is this a special case of a general principle?
- How is this new extended valuation defined?
- Why is it a valuation?
- Is there only such prime ideal $\mathfrak{p}$, so we have a unique extended valuation to choose?
Thanks in advance!