In
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
, the conductor of a
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvabl ...
of
local
Local may refer to:
Geography and transportation
* Local (train), a train serving local traffic demand
* Local, Missouri, a community in the United States
* Local government, a form of public administration, usually the lowest tier of administrat ...
or
global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields:
* Algebraic number field: A finite extension of \mathbb
*Global function fi ...
s provides a quantitative measure of the
ramification in the extension. The definition of the conductor is related to the
Artin map The Artin reciprocity law, which was established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line ...
.
Local conductor
Let ''L''/''K'' be a finite abelian extension of
non-archimedean local field
In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact t ...
s. The conductor of ''L''/''K'', denoted
, is the smallest non-negative
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
''n'' such that the
higher unit group
:
is contained in ''N''
''L''/''K''(''L''
×), where ''N''
''L''/''K'' is
field norm In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield.
Formal definition
Let ''K'' be a field and ''L'' a finite extension (and hence an algebraic extension) of ''K ...
map and
is the
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals cont ...
of ''K''. Equivalently, ''n'' is the smallest integer such that the
local Artin map is trivial on
. Sometimes, the conductor is defined as
where ''n'' is as above.
The conductor of an extension measures the ramification. Qualitatively, the extension is
unramified
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as ...
if, and only if, the conductor is zero, and it is
tamely ramified
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as ...
if, and only if, the conductor is 1. More precisely, the conductor computes the non-triviality of
higher ramification group
In number theory, more specifically in local class field theory, the ramification groups are a Filtration (mathematics), filtration of the Galois group of a local field extension, which gives detailed information on the Ramification (mathematics), ...
s: if ''s'' is the largest integer for which the "
lower numbering" higher ramification group ''G
s'' is non-trivial, then
, where η
''L''/''K'' is the function that translates from "lower numbering" to "
upper numbering" of higher ramification groups.
The conductor of ''L''/''K'' is also related to the
Artin conductor In mathematics, the Artin conductor is a number or ideal associated to a character of a Galois group of a local or global field, introduced by as an expression appearing in the functional equation of an Artin L-function.
Local Artin conductors
...
s of characters of the
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
Gal(''L''/''K''). Specifically,
:
where χ varies over all
multiplicative complex characters of Gal(''L''/''K''),
is the Artin conductor of χ, and lcm is the
least common multiple
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by bot ...
.
More general fields
The conductor can be defined in the same way for ''L''/''K'' a not necessarily abelian finite Galois extension of local fields. However, it only depends on ''L''
ab/''K'', the maximal abelian extension of ''K'' in ''L'', because of the "norm limitation theorem", which states that, in this situation,
:
Additionally, the conductor can be defined when ''L'' and ''K'' are allowed to be slightly more general than local, namely if they are
complete valued field In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size in ...
s with
quasi-finite residue field.
Archimedean fields
Mostly for the sake of global conductors, the conductor of the trivial extension R/R is defined to be 0, and the conductor of the extension C/R is defined to be 1.
Global conductor
Algebraic number fields
The conductor of an abelian extension ''L''/''K'' of number fields can be defined, similarly to the local case, using the Artin map. Specifically, let θ : ''I''
m → Gal(''L''/''K'') be the
global Artin map where the
modulus m is a
defining modulus for ''L''/''K''; we say that
Artin reciprocity
Artin may refer to:
* Artin (name), a surname and given name, including a list of people with the name
** Artin, a variant of Harutyun
Harutyun ( hy, Հարություն and in Western Armenian Յարութիւն) also spelled Haroutioun, Harut ...
holds for m if θ factors through the
ray class group modulo m. We define the conductor of ''L''/''K'', denoted
, to be the highest common factor of all moduli for which reciprocity holds; in fact reciprocity holds for
, so it is the smallest such modulus.
Example
*Taking as base the field of rational numbers, the
Kronecker–Weber theorem
In algebraic number theory, it can be shown that every cyclotomic field is an abelian extension of the rational number field Q, having Galois group of the form (\mathbb Z/n\mathbb Z)^\times. The Kronecker–Weber theorem provides a partial convers ...
states that an algebraic number field ''K'' is abelian over Q if and only if it is a subfield of a
cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of th ...
, where
denotes a primitive ''n''th root of unity. If ''n'' is the smallest integer for which this holds, the conductor of ''K'' is then ''n'' if ''K'' is fixed by complex conjugation and
otherwise.
* Let ''L''/''K'' be
where ''d'' is a
squarefree
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-fr ...
integer. Then,
*:
: where
is the
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
of
.
Relation to local conductors and ramification
The global conductor is the product of local conductors:
:
As a consequence, a finite prime is ramified in ''L''/''K'' if, and only if, it divides
.
An infinite prime ''v'' occurs in the conductor if, and only if, ''v'' is real and becomes complex in ''L''.
Notes
References
*
*
*
*
*
* {{Citation
, last=Serre
, first=Jean-Pierre
, author-link=Jean-Pierre Serre
, chapter=Local class field theory
, title=Algebraic Number Theory, Proceedings of an instructional conference at the University of Sussex, Brighton, 1965
, editor-last=Cassels
, editor-first=J. W. S.
, editor-link=J. W. S. Cassels
, editor2-last=Fröhlich
, editor2-first=Albrecht
, editor2-link=Albrecht Fröhlich
, publisher=Academic Press
, location=London
, isbn=0-12-163251-2
, mr=0220701
, year=1967
Class field theory