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 $ 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 $\sigma^4+\sigma^2+\sigma^2+\sigma+1 =0$ that it exist an ambiguous ideal class??

For example, in the case of $\mathbb{Q}(\sqrt[3]{n},\zeta_3)/\mathbb{Q}(\zeta_3)$ we have $\sigma^3=1$ and $ \sigma^2+\sigma+1=0$ and we have $\mathcal{A}^{3}=\mathcal{A}$, so we prouve that $\mathcal{A}^{\sigma-1}$ is ambigous. I need to do the same for the case of 5