# Understanding problem in Milne's notes on class field theory

I was going through Milne's notes on class field theory and approached the following difficulty in understanding:

On page 155, after the discussion of the conductor, he just writes that any Abelian extension $$L$$ of a number field $$K$$ is contained in some ray class field $$L_\mathfrak{m}$$ for some modulus $$\mathfrak{m}$$. However, I do not see how this follows from the discussion. Can anyone elaborate on this please?

Furthermore, in Example 3.10, also on p. 155, he states that the ray class field of $$\mathbb{Q}$$ is given by $$\mathbb{Q} (\zeta+\zeta^{-1})$$ resp. $$\mathbb{Q}(\zeta)$$ depending on whether the modulus contains an infinite prime. Does anyone know a reference for a computation of this not involving the notions of Ideles?

Here are the notes: https://www.jmilne.org/math/CourseNotes/CFT310.pdf

• Did you mean that the Artin map from $I_K$ to the Galois group of $L/K$ has its kernel contained in $P_{K,1 \bmod^* m} N_{L/K}(I_{L,m})$ for some modulus $m$ ? This is quite the main theorem of CFT. Once you assume it you can take the maximal abelian extension $F/K$ whose kernel of the Artin map is in $P_{K,1 \bmod^* m} N_{F/K}(I_{F,m})$ and $L\subset F$. – reuns Dec 8 '18 at 21:40