real and complex embeddings
Let be a subfield of .
Definition 1.
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1.
A real embedding of is an injective field homomorphism
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2.
A (non-real) complex embedding of is an injective field homomorphism
such that .
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3.
We denote the set of all embeddings, real and complex, of in (note that all of them must fix , since they are field homomorphisms).
Note that if is a real embedding then , where denotes the complex conjugation automorphism:
On the other hand, if is a complex embedding, then is another complex embedding, so the complex embeddings always come in pairs .
Let be another subfield of . Moreover, assume that is finite (this is the dimension of as a vector space over ). We are interested in the embeddings of that fix pointwise, i.e. embeddings such that
Theorem 1.
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 |
---|---|
Canonical name | RealAndComplexEmbeddings |
Date of creation | 2013-03-22 13:54:43 |
Last modified on | 2013-03-22 13:54:43 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 4 |
Author | alozano (2414) |
Entry type | Definition |
Classification | msc 12D99 |
Related topic | GaloisGroup |
Related topic | TotallyRealAndImaginaryFields |
Related topic | RamificationOfArchimedeanPlaces |
Defines | real embedding |
Defines | complex embedding |