Let $l$ be a prime number(even or odd), $n \geq 1$ an interger. Let $\zeta$ be a primitive $l^n$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$.
Is the following proposition true? If yes, how would you prove this?
Proposition The only prime number which ramifies in $K$ is $l$, except $l = 2$ and $n = 1$.
Motivation How a prime number is decomposed in $K$ is fundamental in algebraic number theory. For example, it has a relation with the quadratic reciprocity law.
Effort We consider the case $n = 1$. By the first link below, a prime number $p \neq l$ is unramified. By the third and the fourth link below, $l$ is ramified.
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