Which primes ramify when adjoining roots of a unit?

Question

1) For ANF $K$, if $\zeta_n, u\in\cal O_k^\times$, are there any primes that ramify in $k(\sqrt[n]{u})/k$?

2) is the HCF composed solely of all such extensions, or are there others?

Long story: Let $k$ be an algebraic number field. I am looking for primes that ramify in arbitrary extensions $L/k$. I decompose the extension $L/k$ into smaller extensions, which are easier to analyze:

• A number of cyclotomic extensions of $k$. If $q^n$-th root of unity is adjoined, then $q$ is the only prime of $L$ that can ramify. (Proof: see Ramified primes in a cyclotomic number field of a prime power order)
• A number of roots of non-units of $\cal O$. If we are adjoining $\sqrt[n]{p}$, then the only ramified primes are those dividing the ideal generated by $p$ in $\cal O$ (I think this one's an if-and-only-if), and also over $n$.
• A number of roots of units of $\cal O$. Once again, the primes over $n$ can ramify; how do I find out whether or not they do? [partially answered by Lubin; edited to reflect mercio's answer] Are there others that ramify?
• Also, is the Hilbert Class Field composed chiefly of adjoining roots of units of $\cal O$?

References are fine also.

Answer 1

When you adjoin an $n$th root, primes over $n$ will often ramify. For example consider adding a square root of $3$ to $\Bbb Q$.

Answer 2

1. In the extension $k(\sqrt[n]{u})/k$, only primes dividing $n$ can ramifiy. Whether they do depends on congruences of $u$ modulo powers of prime ideals above $n$. You will find what you need in Hecke's book (Lectures on the Theory of Algebraic Numbers, § 39).

2. If $k$ contains the appropriate roots of unity, then you get the Hilbert class field by adjoining the $n$-th root of an element for which the ideal it generates is an $n$-th power. If $k$ does not have the necessary roots of unity you will have to adjoin them first.