Takagi Existence Theorem - Formulation

Formulation

Here a modulus (or ray divisor) is a formal finite product of the valuations (also called primes or places) of K with positive integer exponents. The archimedean valuations that might appear in a modulus include only those whose completions are the real numbers (not the complex numbers); they may be identified with orderings on K and occur only to exponent one.

The modulus m is a product of a non-archimedean (finite) part m_f and an archimedean (infinite) part m_∞. The non-archimedean part m_f is a nonzero ideal in the ring of integers O_K of K and the archimedean part m_∞ is simply a set of real embeddings of K. Associated to such a modulus m are two groups of fractional ideals. The larger one, I_m, is the group of all fractional ideals relatively prime to m (which means these fractional ideals do not involve any prime ideal appearing in m_f). The smaller one, P_m, is the group of principal fractional ideals (u/v) where u and v are nonzero elements of O_K which are prime to m_f, u ≡ v mod m_f, and u/v > 0 in each of the orderings of m_∞. (It is important here that in P_m, all we require is that some generator of the ideal has the indicated form. If one does, others might not. For instance, taking K to be the rational numbers, the ideal (3) lies in P₄ because (3) = (−3) and −3 fits the necessary conditions. But (3) is not in P_4∞ since here it is required that the positive generator of the ideal is 1 mod 4, which is not so.) For any group H lying between I_m and P_m, the quotient I_m/H is called a generalized ideal class group.

It is these generalized ideal class groups which correspond to abelian extensions of K by the existence theorem, and in fact are the Galois groups of these extensions. That generalized ideal class groups are finite is proved along the same lines of the proof that the usual ideal class group is finite, well in advance of knowing these are Galois groups of finite abelian extensions of the number field.