Hello, this was a problem on an exam. The problem was to generate a field of order 16, and list the elements as powers of a generator (i.e. find an element that would generate the field), and then to list the automorphisms of that field.
I got the first parts by noting that , so I took quotient field of the polynomial ring over (an irreducible polynomial in ). The generator would be a polynomial that is relatively prime to ( I can't remember what exactly I used, but I think it was ). I'm okay with that part.
I had problems getting the automorphisms of the field. There's the obvious identity mapping, but I'm not sure about the others. After doing some reading, I know that if we treat each element as a function , then the map that sends to will be another automorphism. I believe these will be all the automorphisms that fixes the constants (not sure).
In a nutshell, my problems are: 1) Would the automorphisms necessarily be K-automorphisms? (K is some field) If yes, why, and what would K be?; and 2) What are the automorphisms?