Let $R(y) = \sum_{i=0}^{n}a_{i}y^i$ is a $n$ degree polynomial in $K[y]$, where $K$ is any finite extension of $\mathbb{Q}_p$(Field of $p$-adic numbers).
Now $R(y)$ is pure if $a_{0} \neq 0, n \geq 1$, and Newton diagram of $R(y)$ is a straight line.
$\textbf{Theorem 1}:$ If the polynomial $R(y)$ is not pure(so that its Newton diagram consists of two or more straight-line segments necessarily of different slopes), then $R(y)$ factors into two non-constant polynomials in $K[y]$. This means Irreducible polynomials are pure.
My question is how can we say that $R(y)$ is irreducible if the Newton diagram is pure and its slope has some special property? More precisely, how can I prove the following theorem?
$\textbf{Theorem 2 :}$ (Generalized Eisenstein criterion) Suppose R(Y ) is pure, and its Newton diagram has slope $k/n$, where $k$ is an integer relatively prime to $n$. Then $R(Y)$ is irreducible.
$\textbf{Hint}$: If $\alpha$ is a root of $R(y)$ in $\tilde{K}$(Algebraic Closure of $K$), then $v_{p}(\alpha) = k/n$. Hence $K(\alpha)/K$ is a totally ramified extension and has degree $n$, so $R(Y)$ is irreducible. Here $v_{p}$ is the valuation function with respect to prime $p$.
Can anyone help me to understand the hint of the proof or any other proof of Theorem 2?