Crosspost from Math.SE as I did not receive an answer there:


In Lang's book *Elliptic Functions*, he shows how to generate the ray class field with conductor $N$ of an imaginary quadratic number field $k$ using the $j$-invariant of an elliptic curve $A/\mathbb{C}$ with $\mathrm{End}(A)\cong\frak{o}\_\text{$k$}$ and the values of the Weber function at $N$-torsion points of $A$. Namely, Theorem 2 of Chapter 10 reads as follows:



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> Let $A$ be an elliptic curve whose ring of endomorphisms is the ring of algebraic integers $\frak{o}\_\text{$k$}$ in an imaginary quadratic number field $k$, and $A$ is defined over $k(j\_A)$. Let $h$ be the Weber function on $A$, giving the quotient of $A$ by its group of automorphisms. Then $k(j\_A, h(A\_N))$ is the ray class field of $k$ with conductor $N$.
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However, his proof starts out by saying "[l]et $K$ be the smallest Galois extension of $k$ containing $j\_A=j(\frak{a})$ and all coordinates $h(A\_N)$" and then he goes on to prove that $K$ is the ray class field of $k$ with conductor $N$. After that, he concludes that "[t]his proves Theorem 2", without ever mentioning the fact that we have not yet proved $K=k(j\_A, h(A\_N))$, i.e. we do not yet know that $k(j\_A, h(A\_N))$ is Galois. Is there an obvious reason I am missing here?


Notice that he does a similar thing when he proves that the Hilbert class field of $k$ is $k(j(\frak{a}))$ with $\frak{a}$ some fractional ideal of $k$: He starts by defining $K$ as the smallest Galois extension of $k$ containing all $j(\frak{a}\_\text{$i$})$, where the $\frak{a}\_\text{$i$}$ are a set of representatives for the ideal class group, proves that $K$ is the Hilbert class field of $k$ and then decides to be done. However, here I can conclude the argument: It was shown in the course of the proof that all $j(\frak{a}\_\text{$i$})$ are conjugate, hence $j(\frak{a})$ has at least degree $h\_k$ over $k$ and since this is also the degree of the Hilbert class field of $k$ over $k$, the Hilbert class field must already be $k(j(\frak{a}))$. Maybe something similar is possible for the ray class field?



Lang's proof concerning ray class fields of imaginary quadratic number fields
I have found out what's going on here and it is so trivial that I wonder why this did not occur to me earlier: The conclusion of the proof is that $K$ is the ray class field of $k$ modulo $N$ - but this means that $K$ is abelian over $k$, so the intermediate field $k(j\_A, h(A\_N))$ must be Galois over $k$ because all intermediate extensions of abelian extensions are Galois! Duh ...


