This is a problem similar to one of my homework problems, but not on the homework. The problem states that:
Find a primitive root $\beta$ of $F_2[x]/(x^4+x^3+x^2+x+1)$.
I know what a primitive root of a prime number is, but what is a primitive root of a polynomial (or is it called something like a field extension here)?
My book gives this hint: $[x]=\alpha$ doesn't work because $\alpha^5=1$. There are eight choices of $\beta$. I am basically lost on the hint. What is $\alpha$? Why doesn't it work? Why are there 8 choices for $\beta$? Wouldn't there be $2^4 =16$ choices? (or that's the number of polynomials in the field?)
Sorry for the long questions, since I am quite lost right now. Hopefully my questions make sense, and any help would be appreciated!