We try to solve the cyclotomic equation $x^p-1=(x-1)(x^{p-1}+x^{p-2}+...+1)=0$ algebraically. (Transcendentally, we know the roots are $e^{2\pi ik/p}$ for $k=0,...,p-1$.)

It can be easily shown that if $\gcd (m,n)=1$, then a primitive $m$th root of unity times a primitive $n$th root of unity is a primitive $mn$th root of unity, thus we need only consider prime powers. But then if $\alpha$ is a primitive $p$th root of unity, then $\sqrt[k]{\alpha }$ is a primitive $p^k$th root of unity, so we need only consider the case where $p$ is prime.

In general we can use Gauss' method, but let us see how far elementary methods lead us.

$p=3$: we merely solve the quadratic $x^2+x+1=0$ to obtain

$p=5$: we could solve the quartic $x^4+x^3+x^2+x+1=0$ but since it is palindromic we make the variable substitution $y=x+1/x$, and solve

to find

and $x^2-yx+1=0$ implies

giving the four solutions

$p=7$: the palindrome yields a cubic which can be solved for $x$.

$p=11$: the palindrome yields a quintic. Now elementary methods fail us and we need resort to Gauss' method as Vandermonde did.