Cfrgtkky

$text{Newton Polygons for Polynomials}$

There is a lemma in the book $ text{p-adic numbers, p-adic analysis and zet-functions} $ of the author $ text{Neal Koblitz} $ which I mentioned below:

$ text{Lemma 4}: $ Let $ f(X)=(1-frac{X}{alpha_1})(1-frac{X}{alpha_2}) cdots (1-frac{X}{alpha_n})$ be the factorisation of $ f(X)$ in terms of its roots $ alpha_i in Omega$. Let $ lambda_i= text{ord}_p (frac{1}{alpha_i})$. Then, if $ lambda $ is a slope of the Newton polygon having length $l$, it follows that precisely $ l$ of the $ lambda_i$ are equal to $ lambda$.

($Omega$ is a field which is the completion of the algebraic closure of the p-adic field $ mathbb{Q}_p$)

In other words, the slopes of the Newton polygon of $f(X)$ are (counting multiplicity) the p-adic ordinals of the reciprocal roots of $f(X)$.

This is all about the Lemma.

My question-

Is the length $l$ of the Newton Polygon a natural number?

If it is true, then why?

According to the first statement of the Lemma, it says $ text{precisely $l$ of the $ lambda_i$ are equal to $ lambda$}$, which imply $ l$ must be a natural number.

Though I could not explain it why it is a natural number or why should the length of the newton polygon be a natural number.

Please someone explain it.