  This paper presents a connection between Galois points and rational functions
over a finite field with small value sets. This paper proves that the defining
polynomial of any plane curve admitting two Galois points is an irreducible
component of a polynomial obtained as a relation of two rational functions. A
recent result of Bartoli, Borges, and Quoos implies that one of these rational
functions over a finite field has a very small value set, under the assumption
that Galois groups of two Galois points generate the semidrect product. When
two Galois points are external, this paper proves that the defining polynomial
is an irreducible component of a polynomial with separated variables. This
connects the study of Galois points to that of polynomials with small value
sets.
