TITLE: A valuation ring
QUESTION [6 upvotes]: In Qing Liu, Algebraic Geometry and Arithmetic Curves, page 116, exemple 4.1.8, one has $\mathcal{O}_K$ a discrete valuation ring with uniformizing parameter $t$, $P\in\mathcal{O}_K[S]$ an Eisenstein polynomial in $t$ (then irreducible), $\mathcal{O}_L:=\mathcal{O}_K[S]/(P(S))$ (then a integral domain) and with notation $s=\overline{S}$ one has $L=\mathrm{Frac}(\mathcal{O}_L)=K[S]/(P(S))=\bigoplus_{0\leq i\leq n-1} s^i K$.
Then we want to prove that, with $\nu_L\left(\sum_{i=0}^{n-1}\alpha_i s^i\right)=\mathrm{min}\{n\nu_K(\alpha_i)+i,i=0,\ldots,n-1\}$, then $\mathcal{O}_L$ is the discrete valuation ring of $L$.
I'm ok with $\nu_L(x+y)\geq\mathrm{min}\{\nu_L(x),\nu_L(y) \}$.
I'm ok with $\nu_L(x)\geq0\iff x\in\mathcal{O}_L$.
I'm ok with surjectivity onto $\mathbb{Z}$ if I prove the next point.
But for $\nu_L\left(\sum_{i=0}^{n-1}\alpha_i s^i \sum_{j=0}^{n-1}\beta_j s^j\right)=\nu_L\left(\sum_{i=0}^{n-1}\alpha_i s^i\right)+\nu_L\left(\sum_{i=0}^{n-1}\beta_i s^i\right)$ I can't find because as $\nu_L$ is defined only on the form $\sum_{i=0}^{n-1}\alpha_i s^i$ the calcul of $\nu_L\left(\sum_{i=0}^{n-1}\alpha_i s^i \sum_{j=0}^{n-1}\beta_j s^j\right)=\nu_L\left(\sum_k\left(\sum_{i+j=k}\alpha_i+\beta_j\right)s^k\right)$ is too difficult for me (because of the $s^ k$, $k\geq n$).
Maybe with a formula for $\nu_L$ not only with the restricted form $\sum_0^{n-1}\alpha_i s^i$?
Maybe a classical trick for simpllifying the $s^k, k\geq n$?

REPLY [2 votes]: I think it is better to give an equivalent definition of $v_L$: for all $b\in L$, 
$$ v_L(b)=v_K(N_{L/K}(b)).$$
Then use the multiplicativity of the norm $N_{L/K}$. 
Let us prove the equivalence with the definition given in the book. 
Let $b=\sum_{i\le n-1}\alpha_i s^i\in L$. 
Suppose that the minimum is reached at $i_0$. Then 
$$\alpha_is^i=(\alpha_{i_0}s^{i_0})u_is^j, \quad u_i\in \mathcal O_K $$
where $j=i-i_0$ if $i\ge i_0$ and $j=n+(i-i_0)$ if $i<i_0$. So $b=(\alpha_{i_0}s^{i_0})(1+sc)$ with $c\in\mathcal O_L$. As $v_K(N_{L/K}(s))=1$, we are reduced to prove $N_{L/K}(1+sc)$ is a unit. But this is clearly true (either by the very definition of the norm, or compute the norm modulo the maximal ideal of $\mathcal O_K$, or in the separable extension case use the product of the conjugates).