polynomials in finite field: extracting coefficients

edit

Hi i would like to go a bit deeper than this question

http://stackoverflow.com/questions/21...

F.<e> = GF(16)
p = e.minpoly()
p

x^4 + x + 1

R.<x> = PolynomialRing(F)
g=(x+e)*(x+e^14)
g in R

f=-((g(x)-g(e^2))/(x-e^2))*(1/g(e^2))
f in R
f

(e^3 + e^2 + e + 1)*x + e^3 + e

(1) it looks like f is not recognized as a polynomial as it was defined as a rational function which happened to simplify into a polynomial. As such, trying to use a method like f.list() or f.coeff() would cause an error

AttributeError: 'FractionFieldElement_1poly_field' object has no attribute 'degree'

(2) it happens that the coefficient for x is actually equal to e^12, and the constant coefficient is e^9 is there an option when working in GF(16) to display every element as a power of e instead of a linear combination of 1,e,e^2,e^3 ?

thanks