View source for Conductor of an Abelian extension

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Let $L/K$ be an Abelian extension of [[global field]]s and let $N_{L/K} C_L$ be the corresponding subgroup of the idèle class group $C_K$ (cf. [[Class field theory]]). The conductor of an Abelian extension is the greatest common divisor of all positive divisors $n$ such that $L$ is contained in the ray class field $K^n$ (cf  [[Modulus in algebraic number theory]]).

For an Abelian extension of local fields $L/K$ the conductor of $L/K$ is $\mathfrak{p}_K^n$, where $\mathfrak{p}_K$ is the maximal ideal of (the ring of integers $A_K$ of) $K$ and $n$ is the smallest integer such that $N_{L/K} L^* \subset U_K^n = \{ x \in A_K : x \equiv 1 \pmod{\mathfrak{p}_K^n} \}$, $U_K^0 = U_k = A_K^*$. (Thus, an Abelian extension is unramified if and only if its conductor is $A_K$.) The link between the local and global notion of a conductor of an Abelian extension is given by the theorem that the conductor $\mathfrak{f}$ of an Abelian extension $L/K$ of number fields is equal to $\prod_{\mathfrak{p}} \mathfrak{f}_{\mathfrak{p}}$, where $\mathfrak{f}_{\mathfrak{p}}$ is the conductor of the corresponding local extension $L_{\mathfrak{p}} / K_{\mathfrak{p}}$. Here for the infinite primes, $\mathfrak{f}_{\mathfrak{p}} = \mathfrak{p}$ or $1$ according to whether $L_{\mathfrak{p}} \neq K_{\mathfrak{p}}$ or $L_{\mathfrak{p}} = K_{\mathfrak{p}}$.

The conductor ramification theorem of [[Class field theory|class field theory]] says that if $\mathfrak{f}$ is the conductor of a class field $L/K$, then $\mathfrak{f}$ is not divisible by any prime divisor which is unramified for $L/K$ and $\mathfrak{f}$ is divisible by any prime divisor that does ramify for $L/K$ (cf [[Ramification theory of valued fields]]).

If $L/K$ is the cyclic extension of a local field $K$ with finite or algebraically closed residue field defined by a character $\chi$ of degree 1 of $\mathrm{Gal}(K^{\mathrm{s}}/K)$, then the conductor of $L/K$ is equal to $\mathfrak{p}_K^{\mathfrak{f}(\chi)}$, where $\mathfrak{f}(\chi)$ is the Artin conductor of the character $\chi$ (cf. [[Conductor of a character]]). Here $K^{\mathrm{s}}$ is the separable algebraic closure of $K$. There is no such interpretation known for characters of higher degree.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J.-P. Serre,   "Local fields" , Springer  (1979)  (Translated from French)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Neukirch,   "Class field theory" , Springer  (1986)  pp. Chapt. 4, Sect. 8</TD></TR>
</table>