Ex. $2.3.3$ in Algebraic Number Theory by Neukirch is the following:
Let $k$ be a field and $K = k(t)$ the function field in one variable. Show that the valuations $v_{\mathfrak p} $ associated to the prime ideals $\mathfrak p = (p(t))$ of $k[t]$, together with the degree valuation $v_\infty$, are the only valuations of $K$, up to equivalence. What are the residue class fields?
By valuation, I mean the exponential valuation. For instance the 2-valuation on $\Bbb Z$ is $$v(2^sn) = s , (2,n) = 1$$
Take $k = \Bbb Q$ and a non archimedean valuation $v$ on it and extend it to $k[x]$ by $$f(t) = a_0+a_1t+ \dots +a_nt^n \in K, v(f) = \min\{v(a_0),\dots, v(a_n)\}$$ and to $k(x)$ by $$v(f/g) = v(f)-v(g).$$ This is proven to be a valuation earlier on in the chapter. Which of the valuations mentioned in the problem is this equivalent to?
As discussed in the comments, the highlighted part seems to be based on the extra assumption that the valuation is trivial on $k$, $v(k^\times) = \{0\}$. Under this assumption, the result is proved e.g. here (where $k$ is assumed to be finite at first, but what is actually used in the proof is that the value is trivial on $k$).
Without that assumption, the highlighted part is not true. Generalising your example, if you have a nonarchimedean value on $k$, you can assign to $t$ any value $r\in \mathbb{R}$ you like, and set $$v(\sum_{i=0}^n a_i t^i) = \min_{i} (v(a_i) +ir).$$
These valuations and their extensions to power series and Laurent series that converge on certain annuli are of high importance in $p$-adic functional analysis, number theory and representation theory. (Your example is $r=0$; written with multiplicative values, you are looking at polynomials with "$t$ evaluated on the unit circle".)