Conductor (class field theory) - AbsoluteAstronomy.com

Algebraic number theory

Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...

, the conductor of a 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....

of local

Local field

In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...

or global field

Global field

In mathematics, the term global field refers to either of the following:*an algebraic number field, i.e., a finite extension of Q, or*a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq, the field of rational functions...

s provides a quantitative measure of the ramification

Ramification

In mathematics, ramification is a geometric term used for 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign...

in the extension. The definition of the conductor is related to the Artin map.

Local conductor

Let L/K be a finite abelian extension of non-archimedean local fields. The conductor of L/K, denoted

, is the smallest non-negative integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

n such that the higher unit group

is contained in N_L/K(L^×), where N_L/K is field norm

Field norm

In mathematics, the norm is a mapping defined in field theory, to map elements of a larger field into a smaller one.-Formal definitions:1. Let K be a field and L a finite extension of K...

map. Equivalently, n is smallest such that the local Artin map is trivial on

. Sometimes, the conductor is defined as

where n is as above and

is the maximal ideal

Local field

of K.

The conductor of an extension measures the ramification

Ramification

In mathematics, ramification is a geometric term used for 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign...

. Qualitatively, the extension is unramified if, and only if, the conductor is zero, and it is tamely ramified if, and only if, the conductor is 1. More precisely, the conductor computes the non-triviality of higher ramification groups: 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 conductors of characters of the 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...

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

Least common multiple

In arithmetic and number theory, the least common multiple of two integers a and b, usually denoted by LCM, is the smallest positive integer that is a multiple of both a and b...

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 fields with quasi-finiteQuasi-finite field
In mathematics, a quasi-finite field is a generalisation of a finite field. Standard local class field theory usually deals with complete valued fields whose residue field is finite In mathematics, a quasi-finite field is a generalisation of a finite field. Standard local class field theory usually...

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.

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 m is a defining modulus for L/K, then the conductor of L/K, denoted , is the smallest modulusModulus (algebraic number theory)
In mathematics, in the field of algebraic number theory, a modulus is a formal product of places of a global field...

such that θ factors through the ray class group modulo . It can also be defined as the greatest common divisorGreatest common divisor
In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...

of all defining moduli of θ.

Example

Let p be a prime numberPrime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

and let L/K be where d is a squarefree integer. Then,

where is the discriminantDiscriminant of an algebraic number field
In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the algebraic number field...

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.

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