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 $\displaystyle 16 = 2^4$, so I took quotient field of the polynomial ring $\displaystyle Z_2 \[ x \] $ over $\displaystyle x^4+x+1$ (an irreducible polynomial in $\displaystyle Z_2 \[ x \] $). The generator would be a polynomial that is relatively prime to $\displaystyle x^4+x+1$ ( I can't remember what exactly I used, but I think it was $\displaystyle x+1$). 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 $\displaystyle f(x)$, then the map that sends $\displaystyle f(x)$ to $\displaystyle f(x+1)$ 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?

Thanks!