Polynomials taking integer values on primes in a function field

: Let F q [ x ] be the ring of polynomials over a finite field F q and F q ( x ) its quotient field. Let P be the set of primes in F q [ x ] , and let I be the set of all polynomials f over F q ( x ) for which f ( P ) ⊆ F q [ x ] . The existence of a basis for I is established using the notion of characteristic ideal; this shows that I is a free F q [ x ] -module. Through localization, explicit shapes of certain bases for the localization of I are derived, and a well-known procedure is described as to how to obtain explicit forms of some bases of I .