let $L=\mathbb{Q}(\sqrt[5]{n},\zeta_5)$ and $K=\mathbb{Q}(\zeta_5)$ the $5^{th}$ cyclotomic fields, we now that $[L:K] = 5$ and $\textrm{Gal}(L/K) = \langle \sigma \rangle$ so we call $\mathcal{A}$ an ambigous ideal class of the extension $L/K$ if and only if $\mathcal{A}^{\sigma}= \mathcal{A}$.
My question is how to prove using that $\sigma^4+\sigma^3+\sigma^2+\sigma+1 = 0$ that it exist a non trivial ambiguous ideal class??
The ideal class of $O_L$ is represented by a product of unramified prime ideals $$I=\prod_j P_j$$ above distinct prime ideals $\pi_j O_K$ $$\pi_j O_L=\prod_{i=1}^{c_j} P_{i,j} = \prod_{i=0}^{c_j-1} \sigma^i(P_j)$$ (the LHS is principal because $O_K=\Bbb{Z}[\zeta_5]$ is a PID)
and $c_j\ne 1$ implies $P_j$ is comaximal to $\sigma(P_j)$ thus $I=\sigma(I)$ implies each $c_j=1$ ie. $I = (\prod_j \pi_j)$ is principal, thus only the principal ideal class is self conjugate.