This is a question about Newton polygon.
Consider the polynomial $f(x)=a_0+a_1x+\cdots+a_nx^n \in \mathbb Q_p[x]$. Then the following statement follows from Dumas's criterion on irreducibility of $f(x)$:
Statement: $f$ is irreducible over $\mathbb Q_p$ if its Newton polygon consists of one segment which contains no other lattice points on its except the end points.
So having single segment is not sufficient but it is also required that there are no lattice/integer points on that single segment of the Newton polygon of $f$.
Please correct me if I am wrong.
Now if I consider the the polynomial $g(x)=(x-1)(x-2)(x-3) \cdots (x-(p-1))$.
For $p=5$, the polynomial $g$ has vertices $(0,0),(1,2)(2,1),(3,1),(4,0)$, so it seems the Newton polygon has single segment joining the vertices $(0,0)$ and $(4,0)$ and on other vertices lies on this segment. This contradicts the above statement because $g$ is reducible over $\mathbb Q_p$.
Am I missing something with the polynomial $g(x)$ ?