Takagi existence theorem
Encyclopedia
In class field theory
, the Takagi existence theorem states that for any number field K there is a one-to-one inclusion reversing correspondence between the finite abelian extension
s of K (in a fixed algebraic closure
of K) and the generalized ideal class groups defined via a modulus of K.
It is called an existence theorem
because a main burden of the proof is to show the existence of enough abelian extensions of K.
The modulus m is a product of a non-archimedean (finite) part mf and an archimedean (infinite) part m∞. The non-archimedean part mf is a nonzero ideal in the ring of integers
OK 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 ideal
s. The larger one, Im, is the group of all fractional ideals relatively prime to m (which means these fractional ideals do not involve any prime ideal appearing in mf). The smaller one, Pm, is the group of principal fractional ideals (u/v) where u and v are nonzero elements of OK which are prime to mf, u ≡ v mod mf, and u/v > 0 in each of the orderings of m∞. (It is important here that in Pm, 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 P4∞ 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 Im and Pm, the quotient Im/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.
In concrete terms, for abelian extensions L of the rational numbers, this corresponds to the fact that an abelian extension of the rationals lying in one cyclotomic field also lies in infinitely many other cyclotomic fields, and for each such cyclotomic overfield one obtains by Galois theory a subgroup of the Galois group corresponding to the same field L.
In the idelic formulation of class field theory, one obtains a precise one-to-one correspondence between abelian extensions and appropriate groups of ideles, where equivalent generalized ideal class groups in the ideal-theoretic language correspond to the same group of ideles.
of K, and the existence theorem says there exists a unique abelian extension
L/K with Galois group
isomorphic to the ideal class group of K such that L is unramified at all places of K. This extension is called the Hilbert class field
. It was conjectured by David Hilbert
to exist, and existence in this special case was proved by Furtwängler in 1907, before Takagi's general existence theorem.
A further and special property of the Hilbert class field, not true of other abelian extensions of a number field, is that all ideals in a number field become principal in the Hilbert class field. It required Artin
and Furtwängler to prove that principalization occurs.
, who proved it in Japan during the isolated years of World War I
. He presented it at the International Congress 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.
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
, the Takagi existence theorem states that for any number field K there is a one-to-one inclusion reversing correspondence between the finite abelian extension
Abelian extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension. More generally, a Galois extension is called solvable if its Galois group is solvable....
s of K (in a fixed algebraic closure
Algebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....
of K) and the generalized ideal class groups defined via a modulus of K.
It is called an existence theorem
Existence theorem
In mathematics, an existence theorem is a theorem with a statement beginning 'there exist ..', or more generally 'for all x, y, ... there exist ...'. That is, in more formal terms of symbolic logic, it is a theorem with a statement involving the existential quantifier. Many such theorems will not...
because a main burden of the proof is to show the existence of enough abelian extensions of K.
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 mf and an archimedean (infinite) part m∞. The non-archimedean part mf is a nonzero ideal in the ring of integers
Ring of integers
In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...
OK 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 ideal
Fractional ideal
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed...
s. The larger one, Im, is the group of all fractional ideals relatively prime to m (which means these fractional ideals do not involve any prime ideal appearing in mf). The smaller one, Pm, is the group of principal fractional ideals (u/v) where u and v are nonzero elements of OK which are prime to mf, u ≡ v mod mf, and u/v > 0 in each of the orderings of m∞. (It is important here that in Pm, 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 P4∞ 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 Im and Pm, the quotient Im/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
Ideal class group
In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field can be described by a certain group known as an ideal class group...
is finite, well in advance of knowing these are Galois groups of finite abelian extensions of the number field.
A well-defined correspondence
Strictly speaking, the correspondence between finite abelian extensions of K and generalized ideal class groups is not quite one-to-one. Generalized ideal class groups defined relative to different moduli can give rise to the same abelian extension of K, and this is codified a priori in a somewhat complicated equivalence relation on generalized ideal class groups.In concrete terms, for abelian extensions L of the rational numbers, this corresponds to the fact that an abelian extension of the rationals lying in one cyclotomic field also lies in infinitely many other cyclotomic fields, and for each such cyclotomic overfield one obtains by Galois theory a subgroup of the Galois group corresponding to the same field L.
In the idelic formulation of class field theory, one obtains a precise one-to-one correspondence between abelian extensions and appropriate groups of ideles, where equivalent generalized ideal class groups in the ideal-theoretic language correspond to the same group of ideles.
Earlier work
A special case of the existence theorem is when m = 1 and H = P1. In this case the generalized ideal class group is the ideal class groupIdeal class group
In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field can be described by a certain group known as an ideal class group...
of K, and the existence theorem says there exists a unique abelian extension
Abelian extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension. More generally, a Galois extension is called solvable if its Galois group is solvable....
L/K with Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
isomorphic to the ideal class group of K such that L is unramified at all places of K. This extension is called the Hilbert class field
Hilbert class field
In algebraic number theory, the Hilbert class field E of a number field K is the maximal abelian unramified extension of K. Its degree over K equals the class number of K and the Galois group of E over K is canonically isomorphic to the ideal class group of K using Frobenius elements for prime...
. It was conjectured by David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
to exist, and existence in this special case was proved by Furtwängler in 1907, before Takagi's general existence theorem.
A further and special property of the Hilbert class field, not true of other abelian extensions of a number field, is that all ideals in a number field become principal in the Hilbert class field. It required Artin
Emil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...
and Furtwängler to prove that principalization occurs.
History
The existence theorem is due to TakagiTeiji Takagi
Teiji Takagi was a Japanese mathematician, best known for proving the Takagi existence theorem in class field theory....
, who proved it in Japan during the isolated years of World War I
World War I
World War I , which was predominantly called the World War or the Great War from its occurrence until 1939, and the First World War or World War I thereafter, was a major war centred in Europe that began on 28 July 1914 and lasted until 11 November 1918...
. He presented it at the International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
in 1920, leading to the development of the classical theory of class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
during the 1920s. At Hilbert's request, the paper was published in Mathematische Annalen
Mathematische Annalen
Mathematische Annalen is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann...
in 1925.