Let $K$ be a nonarchimedian discrete valued field. Let $f$ be a monic irreducible polynomial in $K[x]$.
Let $w$ be an extended valuation to the splitting field of $f$. (The values of roots of $f$ are equal.)
Fixing the values on roots of $f$ and on$K$, $w$ is unique?
No, that's not true. Consider for example $f=x^2+1\in\mathbb{Q}[x]$ and the extension $K=\mathbb{Q}[i]$: the roots of $f$ have value equal to $0$ for every valuation on $K$, because $1$ has value $0$. On the other side we know exactly which $p$-adic valuations of $\mathbb{Q}$ possess two extensions to $K$. Namely those for which the prime number $p$ splits in the ring $\mathbb{Z}[i]$: these are precisely the primes $p$ which are sums of two squares. For example $5$, $13$ or $17$.