How to construct addition and multiplication table for finite field?

Question

For example, choose $p(X) = X^4 + X + 1$, how to construct addition and multiplication tables for $\mathbb{F}_{2}[X] \mod{X^4 + X + 1} $ ?

Answer 1

p = X^4 + X + 1;
mod[x_] := PolynomialMod[x, p, Modulus -> 2]
elements = mod[Plus @@@ Subsets[X^Range[0, 4]]]

And the tables are:

plustab = Outer[mod@*Plus, elements, elements];
timestab = Outer[mod@*Times, elements, elements];

If you prefer to identify the elements by their index in elements, you can do

lookuptable = AssociationThread[elements, Range[Length[elements]]];
plustabidx = Lookup[lookuptable, #] & /@ plustab;
timestabidx = Lookup[lookuptable, #] & /@ timestab;