Reducing monomials in polynomial rings over finite field using fermat's little theorem using maple.

Question

I am trying to do the following: Given a polynomial ring

Z/3[x_1,x_2,x_3]

and, say I have two polynomials

f1 := x[1]^2.
f2 := 2*x[1].

Their sum should be the following

f1+f2 = 2*x[1]^3.

However, if you reduce the sum using Fermat's little theorem, it should read

f1+f2 = 2*x[1].

Now, in Maple, one can use "Domains" to work in a specific polynomial ring. However, even in at polynomial ring over a finite field, Maple does not do this reduction. Does anyone know how to get maple to do these reductions throughout a worksheet?

Answer 1

If you want to identity polynomials of $\mathbb{F}_3[x]$ giving the same value for $x \in \{0,1,2\}$ then what you need is to look at the quotient ring $\mathbb{F}_3[x]/(x(x-1)(x-2))=\mathbb{F}_3[x]/(x^3-x)$.

More generally to identity polynomials in $\mathbb{F}_q[x]$ giving the same value for $x \in \mathbb{F}_q$ then you want to look at the quotient ring $\mathbb{F}_q[x]/(x^q-x)$