Inverses of Polynomial Functions

Let (K, 3) be a (commutative) topological field. (We do not require that multiplicative inversion be continuous, i.e. 3 is a ring topology. See [l, p. 274] for the definition of the latter.) Throughout this paper, 11 will denote a basic system of neighborhoods of zero for 3. Let P(X) be a polynomial in K[X] of degree »sS2, and let S= {P(a)\a£EK}. We will be concerned with suitably defining a multiple-valued inverse P*~ for P on S, and then considering questions of continuity and uniform continuity for P*". We will be particularly interested in polynomials which are monic and of degree 2, or of the form P(X) =Xn. For P(X) =1", P*~ will be called the nth root function. We will show that the uniform continuity of P*~ is sometimes related to 3 being type V. Indeed, if deg P = 2 and char Kj^l, then P*~ is uniformly continuous if and only if 3 is type V. The hypothesis that char K^2 cannot be eliminated, as will be demonstrated by exhibiting nontrivial topological fields of characteristic 2 which are not type V, but in which the (single-valued) square root function is uniformly continuous. In greater generality, for each prime p, we will exhibit topological fields of characteristic p which are not of type V, but in which the pth root function is uniformly continuous. Finally, we will show that inverses of polynomials need not be continuous at all. Specifically, we will exhibit topological fields of arbitrary characteristic other than 2 in which the square root function is discontinuous.2