Let $K$ be a number field, and let $P\in K[x_0,...,x_n],\;P=\sum_{\gamma}a_{\gamma}x^{\gamma}.$ There is a notion of the height of the polynomial, denoted $h(P)$ which can be described in one of the following two (equivalent) ways:
(1) Let $\mathcal{M}$ be a set of peresentatives of places on $K$. For any $\nu\in \mathcal{M}$ we define $|P|_{\nu}=max_{\gamma}|a_\gamma|_{\nu}$. Then we can define $h(P)=\sum_{\nu}ln|P|_{\nu}$. Note that from the product formula it follows that for any $\lambda\neq0$ in $K$ one has $h(\lambda P)=h(P)$.
(2) Let $a$ be the fractional ideal generated by the coefficients of $P$ in $K$. Then one has (the sum below is on all of the inclusions of $K$ into $\mathbb{C}$)
$h(P)=-lnN(a)+\sum_{\sigma:K\to\mathbb{C}}lnmax_{\gamma}|\sigma(a_\gamma)|$.
So for example, if $K=\mathbb{Q}$ and the coefficients of $P$ are integer and coprime, then one will have $h(P)=max_{\gamma}|a_{\gamma}|$.
My question is: can the last remark be generalized to general number fields? I.E if for instance all the coefficients of $P$ are coprime algebraic integers then we will have $h(P)=max_{\gamma}h(a_{\gamma})$?
Infact, I am looking for something much more general: denote $h'(P)=max_{\gamma}h(a_{\gamma})$. Can we compare $h',h$? My guess, is that in general $h'(P)\geq h(P)$ but that there always exists an element $\lambda \in K$ such that $h'(\lambda P)=h(P)$, as long as the field $K$ is actually the field generated by the coefficients of $P$ (over $\mathbb{Q}$), which amounts to the previous paragraph.
Remark: $h'$ is universal in the sense that it does not depend on which number field $K$ we pick as long as it contains all the coefficients of $P$. However $h$ does depend on $K$ and if $K\subset K'$ then $h_{K'}=[K':K]h_{K}$. This is why I added the remark in the previous paragraph about the field $K$ being generated by (the coefficients of) $P$.
Thank you!