Problem: Let $p(z)=p_o+p_1z+...+p_{n-1}z^{n-1}$ be a polynomial of maximum degree $n-1$. Show that $p(z)$ and $z^n-1$ are coprime if and only if
$$\begin{vmatrix} p_0 & p_{n-1} & ... & p_2 & p_1 \\ p_1 & p_{0} & ... & p_3 & p_2 \\ ... & ... & ... & ... & ... \\ p_{n-2} & p_{n-3} & ... & p_{0} & p_{n-1} \\ p_{n-1} & p_{n-2} & ... & p_{1} & p_{0} \end{vmatrix}\neq 0$$
Solution: Let $ A = \begin{pmatrix} p_0 & p_{n-1} & ... & p_2 & p_1 \\ p_1 & p_{0} & ... & p_3 & p_2 \\ ... & ... & ... & ... & ... \\ p_{n-2} & p_{n-3} & ... & p_{0} & p_{n-1} \\ p_{n-1} & p_{n-2} & ... & p_{1} & p_{0} \end{pmatrix}$
and suppose $det(A)\neq 0$. Hence, we can "eliminate" elements in $A$ and get a triangular matrix $R$ for some $p_i\neq 0$. Let us use this fact to prove that gcd($p(z)$, $z^n-1$)$=1$.
Let $n=1$. That is, the greatest common divisor of $p(z)=p_0$ (constant polynomial) and $z-1$ should be $1$. Since $p_i\neq 0$ this means that $p_0\neq 0$ and the greatest common divisor must be $1$. Let's suppose it's true for $n=k$ and we want to prove it's true for $n=k+1$. This gives us that gcd($p(z)=p_o+p_1z+...+p_{k}z^{k}$, $z^{k+1}$) should be equal to 1. Now I would like to use the induction hypothesis and just say 'done'- But this argument feels kinda vague for me. It don't feel like I have done anything to prove the statement (and I also don't know if this is a valid strategy I have chosen or if this is the best way to prove it).
Oh and now it's time to assume that $p(z)$ and $z^n-1$ are coprime and prove that the determinant in this case can't be 0. If two polynomial are coprime they can, in this case, be written like this $a(z)p(z)+b(z)(z^n-1)=1$. Should I use this somehow to prove it? Or use a similar approach as the first one? Or perhaps a do it in a totally different way?
Would be really glad if someone would like to help me out here. Any hint? suggestions? Have been sitting with this exercise for hours now and I really have to proceed and do some other exercises.
Worth to mention: This is an introductory course in Linear Algebra. Thanks :)