I was wondering if there is a way to prove that the multiplicative of a finite field is cyclic by looking at the character table of such a group. In particular, I was wondering if there is a way to avoid direct mention of classification of finite abelian groups (I am aware of the standard proofs from root counting of polynomials etc.).
My initial thoughts were as follows:
Suppose we somehow manage to find the character table of $\mathbb{F}_q^{\times}$. We know this group is abelian so compare it to the character tables of other abelian groups.
Over $\mathbb{C}$ the character table of a finite abelian group uniquely distinguishes the group, so if we were to find the character table for $\mathbb{F}_q^{\times}$ we would be done.
However the finding of the character table seems like the hard thing to do- we must have to use the field structure somehow. So instead maybe we should look for representations over characteristic $p$ rather than $\mathbb{C}$, but I do not know anything about such representations.
I know somewhere we may be indirectly using the classification of abelian groups in these results, but I think it would be an instructive thing to see it all link up still.
Any thoughts would be appreciated.