ramification of archimedean places
Let be a number field.
Notice that any archimedean place can be extended to an embedding , where is a fixed algebraic closure of (in order to prove this, one uses the fact that is algebraically closed and also Zorn’s Lemma). See also this entry (http://planetmath.org/PlaceAsExtensionOfHomomorphism). In particular, if is a finite extension of then can be extended to an archimidean place of .
Let be a finite Galois extension of number fields and let be a (real or a pair of complex) archimedean place of . Let and be two archimedean places of which extend . Notice that, since is Galois, the image of and are equal, in other words:
so and differ by an element of the Galois group. Similarly, if and are complex embeddings which extend , then there is such that
meaning that either (and thus ) or (and thus ). We are ready now to make the definitions.
The inertia subgroup is nontrivial only when is real, is a complex archimedean place of and is the “complex conjugation” map which has order . Therefore or and ramification of archimedean places occurs if and only if there is a complex place of lying above a real place of .
Suppose first that is a real embedding. Then is injective and implies that is the identity automorphism and would be trivial. So let us assume that is a complex archimedean place and let such that
Therefore, either (which implies that is the identity by the injectivity of ) or . The latter implies that , which is simply complex conjugation:
Finally, since is an extension of , the equation restricts to , thus must be real. ∎
Suppose is an extension of number fields and assume that is a totally imaginary (http://planetmath.org/TotallyRealAndImaginaryFields) number field. Then the extension is unramified at all archimedean places.
Since is totally imaginary none of the embeddings of are real. By the proposition, only real places can ramify. ∎
|Title||ramification of archimedean places|
|Date of creation||2013-03-22 15:07:19|
|Last modified on||2013-03-22 15:07:19|
|Last modified by||alozano (2414)|
|Defines||decomposition and inertia group for archimedean places|