Construct a finite field of order 16. And find a primative element.
The Attempt at a Solution
What I did was find an irreducible polynomial in Z/<2> of degree 4. I used f(x)=x^4+x+1.
Then I took a to be a root of f(x) and set a^4=a+1. Then to make the field I just took powers of a. a is clearly a primitive element.
This seems too easy. Does this indeed produce the field? And does this exact method work for constructing any finite field? And if so doesn't it always give us a primitive element right away?