# Conductor (class field theory)

In algebraic number theory, the **conductor** of a finite abelian extension of local or global fields provides a quantitative measure of the ramification 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 *n* such that the higher unit group

is contained in *N*_{L/K}(*L*^{×}), where *N*_{L/K} is field norm map and is the maximal ideal of *K*.[1] 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.[2]

The conductor of an extension measures the ramification. Qualitatively, the extension is unramified if, and only if, the conductor is zero,[3] and it is tamely ramified if, and only if, the conductor is 1.[4] 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.[5]

The conductor of *L*/*K* is also related to the Artin conductors of characters of the Galois group Gal(*L*/*K*). Specifically,[6]

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.

### 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.[7] 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,[8][9]

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-finite residue field.[10]

### 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.[11]

## 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 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.[12][13][14]

#### Example

- Taking as base the field of rational numbers, the Kronecker–Weber theorem states that an algebraic number field
*K*is abelian over**Q**if and only if it is a subfield of a cyclotomic field , where denotes a primitive*n*th root of unity.[15] 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 integer. Then,[16]

- where is the discriminant of .

## Notes

- Serre 1967, §4.2
- As in Neukirch 1999, definition V.1.6
- Neukirch 1999, proposition V.1.7
- Milne 2008, I.1.9
- Serre 1967, §4.2, proposition 1
- Artin & Tate 2009, corollary to theorem XI.14, p. 100
- As in Serre 1967, §4.2
- Serre 1967, §2.5, proposition 4
- Milne 2008, theorem III.3.5
- As in Artin & Tate 2009, §XI.4. This is the situation in which the formalism of local class field theory works.
- Cohen 2000, definition 3.4.1
- Milne 2008, remark V.3.8
- Janusz 1973, pp. 158,168–169
- Some authors omit infinite places from the conductor, e.g. Neukirch 1999, §VI.6
- Manin, Yu. I.; Panchishkin, A. A. (2007).
*Introduction to Modern Number Theory*. Encyclopaedia of Mathematical Sciences.**49**(Second ed.). pp. 155, 168. ISBN 978-3-540-20364-3. ISSN 0938-0396. Zbl 1079.11002. - Milne 2008, example V.3.11
- For the finite part Neukirch 1999, proposition VI.6.5, and for the infinite part Cohen 2000, definition 3.4.1
- Neukirch 1999, corollary VI.6.6

## References

- Artin, Emil; Tate, John (2009) [1967],
*Class field theory*, American Mathematical Society, ISBN 978-0-8218-4426-7, MR 2467155 - Cohen, Henri (2000),
*Advanced topics in computational number theory*, Graduate Texts in Mathematics,**193**, Springer-Verlag, ISBN 978-0-387-98727-9 - Janusz, Gerald (1973),
*Algebraic Number Fields*, Pure and Applied Mathematics,**55**, Academic Press, ISBN 0-12-380250-4, Zbl 0307.12001 - Milne, James (2008),
*Class field theory*(v4.0 ed.), retrieved 2010-02-22 - Neukirch, Jürgen (1999).
*Algebraic Number Theory*.*Grundlehren der mathematischen Wissenschaften*.**322**. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021. - Serre, Jean-Pierre (1967), "Local class field theory", in Cassels, J. W. S.; Fröhlich, Albrecht (eds.),
*Algebraic Number Theory, Proceedings of an instructional conference at the University of Sussex, Brighton, 1965*, London: Academic Press, ISBN 0-12-163251-2, MR 0220701