Conductor (class field theory)

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]

${\displaystyle N_{L/K}\left(L^{\times }\right)=N_{L^{\text{ab}}/K}\left(\left(L^{\text{ab}}\right)^{\times }\right).}$

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]

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]

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 ${\displaystyle {\mathfrak {f}}(L/K)}$, to be the highest common factor of all moduli for which reciprocity holds; in fact reciprocity holds for ${\displaystyle {\mathfrak {f}}(L/K)}$, so it is the smallest such modulus.[12][13][14]

• 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 ${\displaystyle \mathbf {Q} \left(\zeta _{n}\right)}$, where ${\displaystyle \zeta _{n}}$ denotes a primitive nth 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 ${\displaystyle n\infty }$ otherwise.
• Let L/K be ${\displaystyle \mathbf {Q} \left({\sqrt {d}}\right)/\mathbf {Q} }$ where d is a squarefree integer. Then,[16]

  ${\displaystyle {\mathfrak {f}}\left(\mathbf {Q} \left({\sqrt {d}}\right)/\mathbf {Q} \right)={\begin{cases}\left|\Delta _{\mathbf {Q} \left({\sqrt {d}}\right)}\right|&{\text{for }}d>0\\\infty \left|\Delta _{\mathbf {Q} \left({\sqrt {d}}\right)}\right|&{\text{for }}d<0\end{cases}}}$

where ${\displaystyle \Delta _{\mathbf {Q} ({\sqrt {d}})}}$ is the discriminant of ${\displaystyle \mathbf {Q} \left({\sqrt {d}}\right)/\mathbf {Q} }$.

The global conductor is the product of local conductors:[17]

${\displaystyle {\mathfrak {f}}(L/K)=\prod _{\mathfrak {p}}{\mathfrak {p}}^{{\mathfrak {f}}\left(L_{\mathfrak {p}}/K_{\mathfrak {p}}\right)}.}$

As a consequence, a finite prime is ramified in L/K if, and only if, it divides ${\displaystyle {\mathfrak {f}}(L/K)}$.[18] An infinite prime v occurs in the conductor if, and only if, v is real and becomes complex in L.