2
$\begingroup$

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

$\endgroup$
  • $\begingroup$ 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 $L\subset F$. $\endgroup$ – reuns Dec 8 '18 at 21:40
2
$\begingroup$

The sentence "Every abelian extension..." that you (mis)quote is the last sentence of Remark 5.2 of Chapter V of the notes CFT. It is an immediate consequence of the second sentence "According to the Reciprocity Law..." of the same remark.

The statement in Example 3.10 loc. cit. can be deduced from the description of the ray class group in 1.8(c) of the notes and the description of the Frobenius element in, e.g., ANT 8.18.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.