Books which contain a proof of Kronecker's Jugendtraum

Question

I am looking for books which contain a proof of Kronecker's Jugendtraum i.e. the imaginary quadratic field version of Hilbert's twelfth problem.

Though it seems to be solved even the case for CM field, for my knowledge problem, I want you to introduce a proof it requires up to global class field theory to me.

Answer 1

1) D. Cox, "Primes of the form $x^2+ny^2$", Wiley, 2013: Chap.3 (pp. 181-225) contains a detailed study of the theory of complex multiplication, starting from elliptic functions, modular functions, etc. to establish the "main theorem" of complex multiplication : " Let $\mathscr O$ be an order of an imaginary quadratic field $K$, and $\mathscr I$ a proper fractional $\mathscr O$-ideal. Then the $j$-invariant $j(\mathscr I)$ is an algebraic integer and $K(j(\mathscr I))$ is the ring class field of the order $\mathscr O$. In particular, $K(j(\mathscr O_K))$ is the Hilbert class field of $K$." As a complement, the fields generated by some remarkable singular moduli (e.g. the cubic root of the $j$-function) are computed in §3.

2) Cassels-Fröhlich, "Algebraic number theory", Acad. Press, 1967 : Chap.13 (pp. 292-296) by Serre is a compact (yet complete) determination of the maximal abelian extension of an imaginary quadratic field $K$ using the $j$-invariant.

Answer 2

S. G. Vladut: Kronecker's Jugendtraum and Modular Functions (Studies in the development of modern mathematics), Gordon and Breach publishers, 1991.

Answer 3

The nice paper 'Kronecker's Jugendtraum by Paul Breiding and Detchat Samart' (available on https://silo.tips/download/kronecker-s-jugendtraum-paul-breiding-and-detchat-samart) is a good summary of the proof. I hope you find it useful.