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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.

Community
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Alex
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2 Answers2

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  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.

  • Thanks, I figured this out by now, but +1 for the reference. – Alex Dec 13 '16 at 01:58
  • Err... Sorry, where in that book? I am pretty familiar with all the material in the first 5 chapters; the rest seems to be mostly analytic methods. Alternatively, what happens if, say, $u=1$? – Alex Dec 13 '16 at 06:21
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When you adjoin an $n$th root, primes over $n$ will often ramify. For example consider adding a square root of $3$ to $\Bbb Q$.

mercio
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  • Is there a way to determine whether or not they ramify? Also, can there be any others? – Alex Dec 24 '15 at 14:12
  • It depends on a congruence of what you are taking a root of, modulo something. Also, Hilbert class fields are not as easy when you don't have the corresponding root of unity in the field – mercio Dec 24 '15 at 14:33
  • Sorry, your answer is vague. Modulo what? Root of unity corresponding to what? Maybe you can give a reference instead? – Alex Dec 24 '15 at 14:35
  • Sorry, i am on mobile and it's hard to do anything. I actually remember a related question to each of your questions, one about adding fourth roots in Q(i), one about building a cubic HCF without having a cube root of 1 – mercio Dec 24 '15 at 14:41
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    Well, I think that if the $p$-th roots of unity are in the ground field, adjoining the $p$th root of $a$ will be unramified at the prime $p$ if $a\equiv1\pmod{p^2}$. – Lubin Dec 24 '15 at 14:48