Newton polygons of polynomials composed with Eisenstein polynomials

We construct the field of $p$-adic numbers and discuss some of its properties. We then introduce the concept of a Newton polygon of a polynomial with coefficients in $\Z_p$. We prove a theorem about the Newton polygon of a one segment Newton polygon $f$ composed with an Eisenstein Newton polygon $g$. The theorem states that the Newton polygon of $f \circ g$ is the Newton polygon of $f$ stretched by a factor of the degree of $g$. Finally we discuss a conjecture about the Newton polygon of $f \circ g$ if $f$ has more than one segment. The conjecture states that the Newton polygon of $f \circ g$ will again be stretched by a factor of the degree of $g$. We provide some evidence to support this conclusion.