I was reading Jurgen Neukirch and I came across this in lemma $10.1$ and $10.2$
Let $\zeta_n$ be the $n$th root of unity and $n=l^k$, $l$ being a prime, then $(\lambda)=(1-\zeta_n)$ is a principle ideal and $l\mathcal{O}=(\lambda)^{\phi(n)}$, $\phi$ being the euler totient function. This comes from lemma $10.1$. Then in $10.2$ clearly by applying the fundamental identity, as inertia degree is $1$, $\mathcal{O} /\lambda \mathcal{O} \cong \mathbb{Z}/l\mathbb{Z}$ but then next it says, $\mathcal{O} /\lambda \mathcal{O} \cong \mathbb{Z}/l\mathbb{Z}\Rightarrow \mathcal{O} =\mathbb{Z}+\lambda \mathcal{O} $. Why is this true?
I do not think algebraic number theory is of any use here and it is based on the Isomorphism theorem. Can anyone help?