real and complex embeddings
Let be a subfield of .
On the other hand, if is a complex embedding, then is another complex embedding, so the complex embeddings always come in pairs .
For any embedding of in , there are exactly embeddings of such that they extend . In other words, if is one of them, then
Thus, by taking , there are exactly embeddings of which fix pointwise.
Hence, by the theorem, we know that the order of is . The number is usually decomposed as
where is the number of embeddings which are real, and is the number of embeddings which are complex (non-real). Notice that by the remark above this number is always even, so is an integer.
Remark: Let be an embedding of in . Since is injective, we have , so we can regard as an automorphism of . When is a Galois extension, we can prove that , and hence proving in a different way the fact that
|Title||real and complex embeddings|
|Date of creation||2013-03-22 13:54:43|
|Last modified on||2013-03-22 13:54:43|
|Last modified by||alozano (2414)|