[Math] What are the roots of unity in abelian extensions of imaginary quadratic fields

class-field-theorynt.number-theory

What roots of unity can be contained in the abelian extensions of an imaginary quadratic number field $K = \mathbb{Q}(\sqrt{-d})$? In particular, I would like to know:

  1. Is $K(\zeta_n)/K$ an abelian extension for every $n$?

  2. What are the roots of unity in the ray class field of $K$ with conductor $\mathfrak{c}$?

  3. What are the roots of unity in the ring class field of the order $\mathcal{O} = \mathbb{Z} + f\mathcal{O}_K$ with conductor $f$?

Best Answer

Just as a minor warning: even if the conductor is $1$, there might be nontrivial roots of unity in the class field: take $K = {\mathbb q}(\sqrt{-5}\,)$ and ${\mathfrak c} = (1)$; then the ray class field is the Hilbert class field $K(\sqrt{-1})$, which contains the 4th roots of unity. The roots of unity in the Hilbert class field (i.e. for conductor $1$) lie in the genus class field and can be computed easily.

Any additional roots of unity must come from ramified extensions; a necessary condition for the $p$-th roots of unity to lie in the ray class field must be that the ry class number, which is easily computed, be divisible by $p-1$ (or $(p-1)/2$ if the genus class field contains the quadratic subfield of the $p$-th roots of unity).

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