What is Kronecker's Jugendtraum originally? What was the exact statement of Kronecker's Jugendtraum according to Kronecker himself?
Almost every new idea in algebraic number theory from Kronecker is cited as a progress towards Kronecker's Jugendtraum. The works of Kronecker-Weber, Hecke, Hilbert-Furtwangler-Takagi, Artin, Chevalley, Weil, Tate, Taniyama, Shimura, etc., etc., are all cited. If that were it, all this ado would have been more understandable. What is more confusing is that this has gone on for generation after generation and even now some active number theorists are supposedly still trying to fulfill Kronecker's dream.
So it must be really something if it encompasses so much without any definite notion of stoppage. To clear the haze in mind, can somebody please contribute the original statement according to Kronecker? If this can be in English, that will be even better.
And which part of his ``dream'' was still remaining after Kronecker-Weber? And which part remains after classfield theory?
 A: The source of the words "mein liebster Jugendtraum" is a letter of Kronecker to Dedekind, dated March 15 1880. The German passage is
"Meinen besten Dank für Ihre freundlichen Zeilen vom 12.c.! Ich glaube darin einen willkommenen Anlass finden zu sollen, Ihnen mitzutheilen, dass ich heute die letzte von vielen Schwierigkeiten besiegt zu haben glaube, die dem Abschlusse einer Untersuchung, mit der ich mich in den letzten Monaten wieder eingehender beschäftigt habe, noch entgegenstanden. Es handelt sich um meinen liebsten Jugendtraum, nämlich um den Nachweis, dass die Abel'schen Gleichungen mit Quadratwurzeln rationaler Zahlen durch die Transformations-Gleichungen elliptischer Functionen mit singulären Moduln grade so erschöpft werden, wie die ganzzahligen Abel'schen Gleichungen durch die Kreistheilungsgleichungen."
The English translation is
"Thank you very much for your kind lines of the 12th. I believe they are to give me a welcome occasion to let you know that I believe to have overcome today the last of many difficulties that were still withstanding the completion of an investigation which I had taken up again more intensely in the last few months. It concerns the dearest dream of my youth, to wit, the proof that the
Abelian equations with square roots of rational numbers are exhausted by the transformation equations of elliptic functions with singular moduli exactly in the same way as the rational integral Abelian equations by the cyclotomic equations."
Today, Kronecker's Jugendtraum refers to the problem of generalising the theory of complex multiplication, specifically the achievement of explicitly constructing all abelian extensions of an imaginary quadratic number field by adjoining certain values of certain transcendental functions, to arbitrary number fields. In this generality, the problem is still out of reach.
For a detailed discussion of what Kronecker (and later Hilbert) may or may not have had in mind in formulating the problem (quite vaguely, and with mistakes), see Schappacher's article "On the History of Hilbert’s Twelfth Problem, A Comedy of Errors".
