all the points P0, ..., Pn. Stretch the band upwards, such that the band is stuck on its lower side by some of the points (the points act like nails, partially hammered into the xy plane). The vertices of the Newton polygon are exactly those points. For a neat diagram of this see Ch6 §3 of "Local Fields" by JWS Cassels, LMS Student Texts 3, CUP 1986. It is on p99 of the 1986 paperback edition. Main theorem With the notations in the previous section, the main result concerning the Newton polygon is the following theorem, which states that the valuation of the roots of are entirely determined by its Newton polygon: Let be the slopes of the line segments of the Newton polygon of (as defined above) arranged in increasing order, and let be the corresponding lengths of the line segments projected onto the x-axis (i.e. if we have a line segment stretching between the points and then the length is ). The are distinct; ; if is a root of in , ; for every , the number of roots of whose valuations are equal to (counting multiplicities) is at most , with equality if splits into the product of linear factors over . Corollaries and applications With the notation of the previous sections, we denote, in what follows, by the splitting field of over , and by an extension of to . Newton polygon theorem is often used to show the irreducibility of polynomials, as in the next corollary for example: Suppose that the valuation is discrete and normalized, and that the Newton polynomial of contains only one segment whose slope is and projection on the x-axis is . If , with coprime to , then is irreducible over . In particular, since the Newton polygon of an Eisenstein polynomial consists of a single segment of slope connecting and , Eisenstein criterion follows. Indeed, by the main theorem, if is a root of , If were not irreducible over , then the degree of would be , and there would hold . But this is impossible since with coprime . Another simple corollary is the following: Assume that is Henselian. If the Newton polygon of fulfills for some , then has a root in . Proof: By the main theorem, must have a single root whose valuation is In particular, is separable over . If does not