View source for Unramified character

A character (cf. [[Character of a group|Character of a group]]) of the [[Galois group|Galois group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u0957801.png" /> of a Galois extension of local fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u0957802.png" /> that is trivial on the inertia subgroup. Any unramified character can be regarded as a character of the Galois group of the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u0957803.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u0957804.png" /> is the maximal unramified subfield of the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u0957805.png" />. The unramified characters form a subgroup of the group of all characters. A character of the multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u0957806.png" /> of a local field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u0957807.png" /> that is trivial on the group of units of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u0957808.png" /> is also called unramified. This definition is compatible with the preceding one, because by the fundamental theorem of local [[Class field theory|class field theory]] there is for every Abelian extension of local fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u0957809.png" /> a canonical reciprocity homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u09578010.png" /> that enables one to identify the set of characters of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u09578011.png" /> with a certain subgroup of the character group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u09578012.png" />.

For a Galois extension of global fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u09578013.png" /> a character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u09578014.png" /> of the Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u09578015.png" /> is said to be unramified at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u09578016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u09578017.png" /> if it remains unramified in the above sense under restriction to the decomposition subgroup of any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u09578018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u09578019.png" /> lying over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u09578020.png" />. Similarly, a character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u09578021.png" /> of the idèle class group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u09578022.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u09578023.png" /> is called unramified at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u09578024.png" /> if its restriction to the subgroup of units of the completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u09578025.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u09578026.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u09578027.png" /> is trivial, where the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u09578028.png" /> is imbedded in the standard way in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u09578029.png" />.

From global class field theory it follows that these two definitions of being unramified at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095780/u09578030.png" /> are compatible, as in the local case.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Weil,   "Basic number theory" , Springer  (1974)</TD></TR></table>



====Comments====
See [[Ramified prime ideal|Ramified prime ideal]] and [[Inertial prime number|Inertial prime number]] for the notion of inertia subgroup.