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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c0245701.png" /> be an Abelian extension and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c0245702.png" /> be the corresponding subgroup of the idèle class group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c0245703.png" /> (cf. [[Class field theory|Class field theory]]). The conductor of an Abelian extension is the greatest common divisor of all positive divisors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c0245704.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c0245705.png" /> is contained in the ray class field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c0245706.png" />.
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{{TEX|done}}{{MSC|11R}}
  
For an Abelian extension of local fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c0245707.png" /> the conductor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c0245708.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c0245709.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457010.png" /> is the maximal ideal of (the ring of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457011.png" /> of) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457013.png" /> is the smallest integer such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457015.png" />. (Thus, an Abelian extension is unramified if and only if its conductor is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457016.png" />.) The link between the local and global notion of a conductor of an Abelian extension is given by the theorem that the conductor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457017.png" /> of an Abelian extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457018.png" /> of number fields is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457020.png" /> is the conductor of the corresponding local extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457021.png" />. Here for the infinite primes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457022.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457023.png" /> according to whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457024.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457025.png" />.
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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]]).
  
The conductor ramification theorem of [[Class field theory|class field theory]] says that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457026.png" /> is the conductor of a class field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457027.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457028.png" /> is not divisible by any prime divisor which is unramified for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457030.png" /> is divisible by any prime divisor that does ramify for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457031.png" />.
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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 $L_{\mathfrak{p}} = K_{\mathfrak{p}}$.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457032.png" /> is the cyclic extension of a local field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457033.png" /> with finite or algebraically closed residue field defined by a character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457034.png" /> of degree 1 of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457035.png" />, then the conductor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457036.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457037.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457038.png" /> is the Artin conductor of the character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457039.png" /> (cf. [[Conductor of a character|Conductor of a character]]). Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457040.png" /> is the separable algebraic closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024570/c02457041.png" />. There is no such interpretation known for characters of higher degree.
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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]]).
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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====
 
====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>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  J.-P. Serre,  "Local fields" , Springer  (1979)  (Translated from French)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Neukirch,  "Class field theory" , Springer  (1986)  pp. Chapt. 4, Sect. 8</TD></TR>
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</table>

Latest revision as of 19:42, 7 March 2018

2020 Mathematics Subject Classification: Primary: 11R [MSN][ZBL]

Let $L/K$ be an Abelian extension of global fields 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 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

[a1] J.-P. Serre, "Local fields" , Springer (1979) (Translated from French)
[a2] J. Neukirch, "Class field theory" , Springer (1986) pp. Chapt. 4, Sect. 8
How to Cite This Entry:
Conductor of an Abelian extension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conductor_of_an_Abelian_extension&oldid=16618