[math-fun] Synthesizing functions over finite fields with polynomials

It appears that all functions over GF(2) can be synthesized using only polynomials: E.g., f()=0, f()=1 f(x)=0, f(x)=1, f(x)=x, f(x)=1+x not(x)=1+x, and(x,y)=x*y, or(x,y)=x+y+x*y, etc. Q: Can all functions over _any_ finite field by synthesized using only polynomials, and is there a simple synthesis procedure?