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In [[Algebraic number theory|algebraic number theory]], the '''conductor''' of a finite [[Abelian extension|abelian extension]] of [[Local field|local]] or [[Global field|global field]]s provides a quantitative measure of the [[Ramification (mathematics)|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 <math>\mathfrak{f}(L/K)</math>, is the smallest non-negative [[Integer|integer]] ''n'' such that the higher unit group 
:<math>U^{(n)} = 1 + \mathfrak{m}_K^n = \left\{u\in\mathcal{O}^\times: u \equiv 1\, \left(\operatorname{mod} \mathfrak{m}_K^n\right)\right\}</math>

is contained in ''N''<sub>''L''/''K''</sub>(''L''<sup>×</sup>), where ''N''<sub>''L''/''K''</sub> is [[Field norm|field norm]] map and <math>\mathfrak{m}_K</math> is the [[Local field|maximal ideal]] of ''K''.<ref>{{harvnb|Serre|1967|loc=§4.2}}</ref> Equivalently, ''n'' is the smallest integer such that the local Artin map is trivial on <math>U_K^{(n)}</math>. Sometimes, the conductor is defined as <math>\mathfrak{m}_K^n</math> where ''n'' is as above.<ref>As in {{harvnb|Neukirch|1999|loc=definition V.1.6}}</ref>

The conductor of an extension measures the ramification. Qualitatively, the extension is unramified if, and only if, the conductor is zero,<ref>{{harvnb|Neukirch|1999|loc=proposition V.1.7}}</ref> and it is tamely ramified if, and only if, the conductor is 1.<ref>{{harvnb|Milne|2008|loc=I.1.9}}</ref> 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<sub>s</sub>'' is non-trivial, then <math>\mathfrak{f}(L/K) = \eta_{L/K}(s) + 1</math>, where η<sub>''L''/''K''</sub> is the function that translates from "lower numbering" to "upper numbering" of higher ramification groups.<ref>{{harvnb|Serre|1967|loc=§4.2, proposition 1}}</ref>

The conductor of ''L''/''K'' is also related to the [[Artin conductor]]s of characters of the [[Galois group]] Gal(''L''/''K''). Specifically,<ref>{{harvnb|Artin|Tate|2009|loc=corollary to theorem XI.14, p. 100}}</ref>
:<math>\mathfrak{m}_K^{\mathfrak{f}(L/K)} = \operatorname{lcm}\limits_\chi \mathfrak{m}_K^{\mathfrak{f}_\chi}</math>

where χ varies over all [[Multiplicative character|multiplicative complex characters]] of Gal(''L''/''K''), <math>\mathfrak{f}_\chi</math> is the Artin conductor of χ, and lcm is the [[Least common multiple|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.<ref>As in {{harvnb|Serre|1967|loc=§4.2}}</ref> However, it only depends on ''L''<sup>ab</sup>/''K'', the maximal abelian extension of ''K'' in ''L'', because of the "norm limitation theorem", which states that, in this situation,<ref>{{harvnb|Serre|1967|loc=§2.5, proposition 4}}</ref><ref>{{harvnb|Milne|2008|loc=theorem III.3.5}}</ref>
: <math>N_{L/K}\left(L^\times\right) = N_{L^{\text{ab}}/K} \left(\left(L^{\text{ab}}\right)^\times \right).</math>

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 field|quasi-finite]] residue field.<ref>As in {{harvnb|Artin|Tate|2009|loc=§XI.4}}. This is the situation in which the formalism of [[Local class field theory|local class field theory]] works.</ref>

===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.<ref>{{harvnb|Cohen|2000|loc=definition 3.4.1}}</ref>

==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''<sup>'''m'''</sup> → Gal(''L''/''K'') be the global Artin map where the [[Modulus (algebraic number theory)|modulus]] '''m''' is a defining modulus for ''L''/''K''; we say that [[Artin reciprocity law|Artin reciprocity]] holds for '''m''' if θ factors through the ray class group modulo '''m'''.  We define the conductor of ''L''/''K'', denoted <math>\mathfrak{f}(L/K)</math>, to be the highest common factor of all moduli for which reciprocity holds; in fact reciprocity holds for <math>\mathfrak{f}(L/K)</math>, so it is the smallest such modulus.<ref>{{harvnb|Milne|2008|loc=remark V.3.8}}</ref><ref>{{harvnb|Janusz|1973|pp=158,168–169}}</ref><ref>Some authors omit infinite places from the conductor, e.g. {{harvnb|Neukirch|1999|loc=§VI.6}}</ref>

====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|cyclotomic field]] <math>\mathbf{Q}\left(\zeta_n\right)</math>, where <math>\zeta_n</math> denotes a primitive ''n''th root of unity.<ref>{{cite book | first1=Yu. I. | last1=Manin |  first2=A. A. | last2=Panchishkin | title=Introduction to Modern Number Theory | series=Encyclopaedia of Mathematical Sciences | volume=49 | edition=Second | year=2007 | isbn=978-3-540-20364-3 | issn=0938-0396 | zbl=1079.11002 | pages=155, 168 }}</ref>  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 <math>n \infty</math> otherwise.
* Let ''L''/''K'' be <math>\mathbf{Q}\left(\sqrt{d}\right)/\mathbf{Q}</math> where ''d'' is a squarefree integer. Then,<ref>{{harvnb|Milne|2008|loc=example V.3.11}}</ref>
*: <math>\mathfrak{f}\left(\mathbf{Q}\left(\sqrt{d}\right)/\mathbf{Q}\right) = \begin{cases}
        \left|\Delta_{\mathbf{Q}\left(\sqrt{d}\right)}\right| & \text{for }d > 0 \\
  \infty\left|\Delta_{\mathbf{Q}\left(\sqrt{d}\right)}\right| & \text{for }d < 0
\end{cases}</math>
: where <math>\Delta_{\mathbf{Q}(\sqrt{d})}</math> is the [[Discriminant of an algebraic number field|discriminant]] of <math>\mathbf{Q}\left(\sqrt{d}\right)/\mathbf{Q}</math>.

====Relation to local conductors and ramification====
The global conductor is the product of local conductors:<ref>For the finite part {{harvnb|Neukirch|1999|loc=proposition VI.6.5}}, and for the infinite part {{harvnb|Cohen|2000|loc=definition 3.4.1}}</ref>
:<math>\mathfrak{f}(L/K) = \prod_\mathfrak{p}\mathfrak{p}^{\mathfrak{f}\left(L_\mathfrak{p}/K_\mathfrak{p}\right)}.</math>

As a consequence, a finite prime is ramified in ''L''/''K'' if, and only if, it divides <math>\mathfrak{f}(L/K)</math>.<ref>{{harvnb|Neukirch|1999|loc=corollary VI.6.6}}</ref> An infinite prime ''v'' occurs in the conductor if, and only if, ''v'' is real and becomes complex in ''L''.

==Notes==
{{reflist|2}}

==References==
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| first2=John
| title=Class field theory
| publisher=American Mathematical Society
| year=2009
| origyear=1967
| isbn=978-0-8218-4426-7
| mr=2467155
}}
*{{Citation
| last=Cohen
| first=Henri
| year=2000
| title=Advanced topics in computational number theory
| isbn=978-0-387-98727-9
| publisher=[[Physics:Springer-Verlag|Springer-Verlag]]
| series=[[Graduate Texts in Mathematics]]
| volume=193
}}
*{{citation | first=Gerald | last=Janusz | title=Algebraic Number Fields | year=1973 | publisher=Academic Press | isbn=0-12-380250-4 | series=Pure and Applied Mathematics | volume=55 | zbl=0307.12001 }}
*{{Citation
| last=Milne
| first=James
| title=Class field theory
| url=http://jmilne.org/math/CourseNotes/cft.html
| edition=v4.0
| year=2008
| accessdate=2010-02-22
}}
* {{Neukirch ANT}}
* {{Citation
| last=Serre
| first=Jean-Pierre
| 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.
| editor2-last=Fröhlich
| editor2-first=Albrecht
| publisher=Academic Press
| location=London
| isbn=0-12-163251-2 
| mr=0220701
| year=1967
}}

[[Category:Class field theory]]

{{Sourceattribution|Conductor (class field theory)}}