Unramified character

A character (cf. Character of a group) of the Galois group $ G ( K / k ) $ of a Galois extension of local fields $ K / k $ that is trivial on the inertia subgroup. Any unramified character can be regarded as a character of the Galois group of the extension $ K _ { \mathop{\rm unr} } / k $, where $ K _ { \mathop{\rm unr} } $ is the maximal unramified subfield of the extension $ K / k $. The unramified characters form a subgroup of the group of all characters. A character of the multiplicative group $ k ^ {*} $ of a local field $ k $ that is trivial on the group of units of $ k $ is also called unramified. This definition is compatible with the preceding one, because by the fundamental theorem of local class field theory there is for every Abelian extension of local fields $ K / k $ a canonical reciprocity homomorphism $ \theta : k ^ {*} \rightarrow G ( K / k ) $ that enables one to identify the set of characters of the group $ G ( K / k ) $ with a certain subgroup of the character group of $ k ^ {*} $.

For a Galois extension of global fields $ K / k $ a character $ \chi $ of the Galois group $ G ( K / k ) $ is said to be unramified at a point $ \mathfrak Y $ of $ k $ if it remains unramified in the above sense under restriction to the decomposition subgroup of any point $ \mathfrak P $ of $ K $ lying over $ \mathfrak Y $. Similarly, a character $ \chi $ of the idèle class group $ C ( k) $ of $ k $ is called unramified at $ \mathfrak Y $ if its restriction to the subgroup of units of the completion $ k _ {\mathfrak Y } $ of $ k $ relative to $ \mathfrak Y $ is trivial, where the group $ k _ {\mathfrak Y } ^ {*} $ is imbedded in the standard way in $ C ( k) $.

From global class field theory it follows that these two definitions of being unramified at a point $ \mathfrak Y $ are compatible, as in the local case.

References

[1]	A. Weil, "Basic number theory" , Springer (1974)

Comments

See Ramified prime ideal and Inertial prime number for the notion of inertia subgroup.