since we have a finite field, its multiplicative group is cyclic. what are the automorphisms of Z_{15}? (hint: there are 8 of them).

if you call the generator x+1 + (x^{4}+ x + 1), b, then these automorphisms are of the form b→b^{n}, 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 Z_{2}, 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 (a_{1}+ a_{2}b + a_{3}b^{2}+ a_{4}b^{3})^{n}= (a_{1})^{n}+ (a_{2}b)^{n}+ (a_{3}b^{2})^{n}= (a_{4}b^{3})^{n}?