# Class field theory of imaginary quadratic fields

Can someone give me a good source that deals in detail with the Class Field Theory of imaginary quadratic fields?

The sources on CFT that I have at hand only deal with CFT in general and then proceed to do some of the well-known results over $\mathbb {Q}$ like the Kronecker-Weber theorem.

To be precise: I am dealing with $K=\mathbb {Q}(\sqrt {(-d)})$ where $d\geq 2$ is a square-free integer, its Hilbert class field $K_1$, a prime $l$ of $\mathbb {Q}$ that stays inert in $K$. Let $K_l$ be the class field of conductor $l$ of $K$. The authors of one of the papers I am studying for my master's thesis claim that $\text {Gal } \left ( K_l / K_1 \right )$ is cyclic of order $l + 1$.

The authors give no reference on that fact, so I assume that it must be obvious for an expert, which I am not, so please have mercy with me.

• Cox, Primes of the Form $x^2 + n y^2$ – Will Jagy Jul 4 '15 at 20:07
• plentiful worked examples in Hudson and Williams 1991, then Liu and Williams 1994 – Will Jagy Jul 4 '15 at 20:14
• Williams, Kenneth S.; Hudson, Richard H., Representation of primes by the principal form of discriminant $-D$ when the class number $h(-D)$ is $3$. Acta Arith. 57 (1991), no. 2, 131--153. – Will Jagy Jul 4 '15 at 20:15
• K. S. Williams, D. Liu, Representation of primes by the principal form of negative discriminant ∆ when h(∆) is 4, Tamkang J. Math. 25 (1994) – Will Jagy Jul 4 '15 at 20:17
• wiley.com/WileyCDA/WileyTitle/productCd-1118390180.html – Will Jagy Jul 4 '15 at 20:21

The ray class field $K_{\mathfrak m}$ over $K$ of conductor $\mathfrak m$ has Galois group isomorphic to the ray class group $\mathrm{Cl}_{\mathfrak m}$ of the same conductor. This ray class group sits in a short exact sequence $$0 \to (\mathcal O_K/\mathfrak m)^{\times}/\mathcal O_K^{\times} \to \mathrm{Cl}_{\mathfrak m} \to \mathrm{Cl}_1 \to 0.$$ Thus we see that the Galois group $\mathrm{Gal}(K_{\mathfrak m}/K_1)$ is isomorphic to $(\mathcal O_K/\mathfrak m)^{\times}/\mathcal O_K^{\times}$.
If $\mathfrak m = l$ is an inert prime, then this quotient can be identified with $\mathbb F_{l^2}^{\times}/\{\pm 1\}$ (here I'm assuming that $K = \mathbb Q(\sqrt{-d})$ with $d \neq 1$ or $3$, so that $\mathcal O_K^{\times} = \pm 1$), which is cyclic of order $(l^2 - 1)/2$ (assuming $l$ is odd, so that $\pm 1$ really are two distinct elements of $\mathbb F_{l^2}^{\times}$), or of order $3$ (if $l = 2$).
So $l+1$ is a divisor of the order of $\mathrm{Gal}(K_l/K_1)$, but is not equal to its order unless $l = 2$ or $3$.