Let $K$ be a number field. I know what the norm of an element $\alpha \in K$ is. However I have recently learned that one can also define the norm of a polynomial $P \in K[x]$, but I am not sure exactly how one could go about doing this. I would have started by defining the norm of an irreducible polynomial $P$ to simply be the norm of one of its roots in the respective field, but I have also been told that it should satisfy the following property:
For any $P \in K[x]$, there exists a polynomial with integer coefficients which is proportional to $N_{K / \mathbb Q}(P)$. $\tag{P1}$
So this tells me that the norm $N_{K / \mathbb Q}(P)$ that I am looking for should itself be a polynomial which suggested that I look for norms in function fields. Unfortunately, I don't know much about function fields, and some extensive search has only revealed something known as the "norm form". Though I am not very clear on what it means, I have only come to understand that this is a norm defined on linear forms in indeterminates, dependent on the choice of a vector-space basis of finite extensions, so I am still not sure how (if at all) this can be used to give the definition I am looking for.
In short my questions are the following:
(1) Can one define a norm on $K[x]$ (for a finite extension $K/\mathbb Q$, or more generally, for $L/K$ finite and separable) such that the property $(P1)$ holds?
(2) What is the intuition or motivation behind such a definition?
(3) How does it relate to the usual field norm in a finite extension (in particular, is it a function theoretic analogue of the field norm, as in it satisfies complete multiplicativity and there is some nice degree-reduction formula like the relation $N_{L/K}(\alpha) = N_{K(\alpha)/K}(\alpha)^{[L:K(\alpha)]}$, which is valid for the usual field norm)?
(4) Is that norm a well-studied quantity, perhaps with a proper name? (I would appreciate some references as well, since despite my best attempts, I couldn't find any.)
I would really appreciate some suggestions or help.