How do I express each element in a field F as a power of a primitive element? I have a field $\mathbb F_{2^4}$, and it is represented as a residue ring of the

polynomials over $\mathbb F_2$ modulo the polynomial $\beta^4 + \beta^3 + \beta^2 + \beta + 1$.

I want to express each element in this field as a power of a primitive element $β+1$.

My questions are:


What are the elements of the fields?
How do I express them as powers of the given primitive element?


Any hints will be very helpful.
Just enumerate all powers of your primitive element.  There are only 15 of them.  That will give you a lookup table that allows you to translate from field element to its expression as a power of the primitive element.