The Abhyankar- Moh theorem

In this note we present a proof of the following theorem. Theorem 0.1. Let k[t] be the ring of polynomials in one variable t over a field k of charac-teristic zero, and let f, g ∈ k[t]r k. If k[f, g] = k[t], then deg f | deg g or deg g | deg f. The above theorem appeared in the fifties with an incorrect proof in Segre’s paper [12] which discusses a subject of the Jacobian conjecture. The first time it was proved in 1975 by Abhyankar and Moh [3] (see also [1]). The history of this theorem and its applications are presented in the papers [3] and [4]. There appeared also other proofs in the years 1977 – 1982. They were publicated by Miyanishi [7] (see also [8]), Ganong [5] and Rudolph [11]. Abhyankar and Moh proved this theorem basing on the theory of approximate roots of polynomials which was developed by them. In 1986 Richman [10] presented a new interesting proof which does not use approximate roots. There is however a certain gap in his proof, noticed and corrected, in 1991, by Kang [6] (see also [9]). The purpose of this article is a presentation of the full proof of Theorem 0.1 according to Richman. We assumed in Theorem 0.1 that k is a field of characteristic zero. With a certain additional assumption which is ”char(k)- gcd(deg f,deg g)”, this theorem is also true for fields of positive characteristics (see for example [3]). In such version Richman proves it in [10]. But we do not take up this case here. We stress that it is easy to prove Theorem 0.1 when the degree of f or g is a prime numbe