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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.

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I would imagine that you can mimic the proof of the Kronecker-Weber theorem. –  Jonah Sinick Mar 31 '13 at 21:19
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...because the class number is 1, and the unit group is finite. –  Jonah Sinick Mar 31 '13 at 23:07
    
A problem vs KW, maybe small: all non-CFT proofs of KW use the global fact that the cyclotomic Galois group is the product of its ramification groups, as large as possible from the local point of view. CFT says that this is true for no other field. One must show that certain kinds of ramification at one prime are globally possible only with ramification at other primes. Perhaps this is generally easy, but since the unit group is finite, it probably yields to brute force. –  Ben Wieland Apr 15 '13 at 14:24

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up vote 12 down vote accepted

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.

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Thank you, this is great! –  David Corwin Jul 18 '13 at 4:10

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