since we have a finite field, its multiplicative group is cyclic. what are the automorphisms of Z15? (hint: there are 8 of them).
if you call the generator x+1 + (x4 + x + 1), b, then these automorphisms are of the form b→bn, for some n. which n are possible?
EDIT: not all automorphisms of the multiplicative group, will yield field automorphisms. we know that the order of the galois group is the degree of our field of 16 elements over Z2, namely 4.
so only 4 of the 8 multiplicative automorphisms are additive. can you say which ones (think: frobenius maps)?
that is: for which n do we have (a1 + a2b + a3b2 + a4b3)n = (a1)n + (a2b)n + (a3b2)n = (a4b3)n?