Prime Rational Functions and Integral Polynomials

Let f(x) be a complex rational function. In this work, we study conditions under which f(x) cannot be written as the composition of two rational functions which are not units under the operation of function composition. In this case, we say that f(x) is prime. We give sufficient conditions for complex rational functions to be prime in terms of their degrees and their critical values, and we derive some conditions for the case of complex polynomials. We consider also the divisibility of integral polynomials, and we present a generalization of a theorem of Nieto. We show that if f(x) and g(x) are integral polynomials such that the content of g divides the content of f and g(n) divides f(n) for an integer n whose absolute value is larger than a certain bound, then g(x) divides f(x) in Z[x]. In addition, given an integral polynomial f(x), we provide a method to determine if f is irreducible over Z, and if not, find one of its divisors in Z[x].

Calculations on a piece of “Long Point, Head Quarters” stationary with the “Long Point Company, Canada” seal

Calculations on a piece of “Long Point, Head Quarters” stationary with the “Long Point Company, Canada” seal. There are land calculations on this sheet, n.d.

Partial piece of post office registration to J.M. Ball of Toronto, Ont.

Partial piece of post office registration to J.M. Ball of Toronto, Ont. Half of ticket is missing. Text is affected, Jul. 5, 1875.