Takagi existence theorem

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In class field theory, the Takagi existence theorem states in part that if K is a number field with class group G, there exists a unique abelian extension L/K with Galois group G, such that every ideal in K becomes principal in L, and that L is characterized by the property that it is the maximal unramified abelian extension of K. The theorem tells us that the Hilbert class field conjectured by Hilbert always exists, but it required Artin and Furtwängler to prove that principalization occurs.

More generally, the existence theorem tells us that there is a one-to-one inclusion reversing correspondence between the abelian extensions of K and the ideal groups defined via a modulus of K. Here a modulus (or ray divisor) is a formal product of the valuations (also called primes or places) on K to positive integer exponents. The archimedean valuations include only those whose completions are the real numbers; they may be identified with orderings on K and occur only to exponent one.

The modulus μ is a product of an archimedean part α and a non-archimedean part η, and η can be identified with an ideal in the ring of integers O_K of K. The number group mod η of K, K_η, is the multiplicative group of fractions u/v with non-zero u and v prime to η in O_K. The ray or unit ray number group mod μ of K, K_μ¹, adds to the conditions on u and v that u ≡ v mod η and u/v > 0 in each of the orderings of α. A ray number group is now a group lying between K_η and K_μ¹, and the ideal groups mod μ are the fractional ideals prime to η modulo such a ray number group. It is these ideal groups which correspond to abelian extensions by the existence theorem.

The theorem is due to Teiji Takagi, who proved it during the isolated years of World War I and presented at the International Conference of Mathematicians in 1920, leading to the development of the classical theory of class field theory during the 1920s. At Hilbert's request, the paper was published in Mathematische Annalen in 1925.