How to prove that $ (\zeta\_{p^{n+1}}-1)^{p} $ = $ (\zeta\_{p^{n}}-1) $ as ideals where $ \zeta\_{n} $ is a primitive nth root of unity ?



Cyclotomic fields
Suppose $\zeta \_{p^{n+1}}=x$ say.Then, for every $\omega $ a $p$-th root of unity, $x\omega-1=x^r-1$ (for some $r$) generates the same ideal as $x-1$ (clearly, it is contained in the ideal generated by $x-1$; by switching $x$ and $x\omega$ we get the other statement). 


Taking the product over all the $\omega $ we now get the result you wanted. 


