belong to , has a distinct Galois conjugate over , with , and is a root of , a contradiction. More generally, the following factorization theorem holds: Assume that is Henselian. Then , where , is monic for every , the roots of are of valuation , and . Moreover, , and if is coprime to , is irreducible over . Proof: For every , denote by the product of the monomials such that is a root of and . We also denote the factorization of in into prime monic factors Let be a root of . We can assume that is the minimal polynomial of over . If is a root of , there exists a K-automorphism of that sends to , and we have since is Henselian. Therefore is also a root of . Moreover, every root of of multiplicity is clearly a root of of multiplicity , since repeated roots share obviously the same valuation. This shows that divides Let . Choose a root of . Notice that the roots of are distinct from the roots of . Repeat the previous argument with the minimal polynomial of over , assumed w.l.g. to be , to show that divides . Continuing this process until all the roots of are exhausted, one eventually arrives to , with . This shows that , monic. But the are coprime since their roots have distinct valuations. Hence clearly , showing the main contention. The fact that follows from the main theorem, and so does the fact that , by remarking that the Newton polygon of can have only one segment joining to . The condition for the irreducibility of follows from the corollary above. (q.e.d.) The following is an immediate corollary of the factorization above, and constitutes a test for the reducibility of polynomials over Henselian fields: Assume that is Henselian. If the Newton polygon does not reduce to a single segment then is reducible over . Other applications of the Newton polygon comes from the fact that a Newton Polygon is sometimes a special case of a Newton polytope, and can be used to construct asymptotic solutions of two-variable polynomial equations like Symmetric function explanation In the context of a valuation, we are given certain information in the form of the valuations of elementary symmetric functions of the roots of a polynomial, and require information on the valuations of the actual