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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.

Related question:

Decomposition of a prime number in a cyclotomic field

On the ring generated by an algebraic integer over the ring of rational integer

Special units of the cyclotomic number field of an odd prime order $l$

Decomposition of $l$ in a subfield of a cyclotomic number field of an odd prime order $l$

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    Yes, the result is correct. Refer to _Algebraic Number Theory_ by Neukirch page 61-63.2012-07-26
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    This is definitely done in [Jim Milne's notes](http://jmilne.org/math/CourseNotes/ant.html), and it should be in any book on algebraic number theory. The proof I've seen uses the third link you gave.2012-07-26
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    @DylanMoreland The third link can be used only for the proof that $l$ is ramified. It cannot be used to show that other prime numbers are unramified.2012-07-26
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    @MakotoKato Sure. I guess I don't see that as being the hard part — $X^{l^n} - 1$ is, after all, separable modulo the other primes.2012-07-26
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    @DylanMorland You know there are several answers to a question. I guess Milne's proof is not the only one. It's nice to have several proofs, here.2012-07-27
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    Please let me know the reason for the downvotes. Unless you make it clear, it's hard to improve my question.2012-07-28

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