In Noah Snyder's historical undergraduate thesis on Artin L-Functions, it mentions that Takagi proved Kronecker's Jugendtraum in the case of Q(i) in his doctoral thesis. Since I don't know how to get access to Takagi's thesis, does anyone have any idea how he did this?


I know how to prove complex multiplication (e.g. as in Silverman's book) by first assuming class field theory. But Takagi was working before class field theory had been demonstrated. So how did he do it?


I understand how one can directly prove the reciprocity law directly for extensions generated by torsion points on elliptic curve. The nontrivial part is this: how does one prove that an *arbitrary* abelian extension of an imaginary quadratic field arises in such a manner?


Note that this question has everything to do with my [previous question](https://mathoverflow.net/questions/100276/can-one-prove-complex-multiplication-without-assuming-cft).



How did Takagi prove Kronecker's Jugendtraum for Q(i)?
Takagi's goal is the following: 


1. show the existence of sufficiently many cyclic extensions defined 
 by division values of sn $u$ (the lemniscatic sine);
2. prove that each abelian extension of ${\mathbb Q}(i)$ is contained in 
 the compositum of these fields.


Step 1 is analogous to the construction of the fields of $p^n$-th roots
of unity (in particular, proving the irreducibility of the corresponding
cyclotomic polynomials and determining their dirscriminants), whereas
Step 2 is the analogue of the theorem of Kronecker-Weber.


Step 1


Takagi (thesis, Sect. 6) constructs the following cyclic extensions
of ${\mathbb Q}(i)$:


* For each Gaussian prime $\mu$ such that $p^h$ is the largest power
 of the prime $p$ that divides $(N\mu-1)$, there exists a cyclic 
 extension unramified outside $\mu$ with degree $p^h$.
* For each Gaussian prime $\pi$ with odd prime norm $p \equiv 1 \bmod 4$
 and each integer $\lambda \ge 1$ there exists a cyclic extension of 
 prime power degree $p^\lambda$ unramified outside $\pi$.
* For each prime number $q \equiv 3 \bmod 4$ and each integer 
 $\lambda \ge 1$ there exist $q^\lambda +1$ cyclic extensions with
 degree $q^\lambda$ unramified outside $q$.
* For each Gaussian prime $\mu$ with $N\mu-1 = 2^{h+2} u$, where
 $u$ is an odd integer, there is a cyclic extension of degree $2^{h+2}$
 unramified outside $(1+i)\mu$ for which the subfield of degree $2^h$ 
 is unramified outside $\mu$.
* For each integer $\lambda > 1$ there are $2^\lambda +1$ cyclic
 extensions of degree $2^\lambda$ unramified outside $1+i$.


Takagi also proves the decomposition law in these extensions.


Step 2


In Sect. 9, Takagi begins proving the analogue of the 
theorem of Kronecker-Weber. His approach is the one used by Hilbert
in his proof of the theorem of Kronecker-Weber: Given an abelian
extension $L/{\mathbb Q}(i)$, we write $L$ as a compositum of cyclic 
extensions of prime power degree. If the odd Gaussian prime $\mu$ is 
ramified in $L$, form the compositum of $L$ and the cyclic extension 
$K/{\mathbb Q}(i)$ unramified outside $\mu$ constructed in Step 1 and 
show that the compositum $KL$ contains a cyclic extension 
$M/{\mathbb Q}(i)$ unramified at $\mu$ such that $L$ is contained 
in $KM$; the case of wild ramification requires a careful investiagtion 
of the ramification subgroups. 


By induction, this process reduces the proof to cyclic extensions that 
are unramified everywhere. There are various ways of proving that 
such extensions do not exist:


1. We can use Takagi's Lemma, according to which every cyclic extension 
of ${\mathbb Q}(i)$ that is normal over ${\mathbb Q}$ is a cyclotomic 
field. This is how Takagi proves this claim.
2. By Hilbert's Theorem 94, unramified cyclic extensions of prime
degree $\ell$ of a number field $K$ exist only if $K$ has class
number divisible by $\ell$.
3. Minkowski's bounds show that ${\mathbb Q}(i)$ does not 
admit any nontrivial unramified extension.


P.S. Takagi's thesis is contained in his collected papers. I also would like to refer you to a beautiful article by Cox and Hyde on [The Galois theory of the lemniscate](http://arxiv.org/abs/1208.2653).


