First, if I may, I would like to ask for help in getting an intuitive understanding regarding embeddings.
Wikipedia gives examples such as $\mathbb N$ in $\mathbb Z$. My first question is: were they not there already? And my second question is what advantage (or advantageous constraint) do you get by embedding.
This general question is motivated by a Theorem in Appendix 2 in Marcus's "Number Fields" on p. 259.
He states: We are interested in the embeddings of L in $\mathbb C$ which fix K pointwise. (L and K are subfields of $\mathbb C$ and K $\subset$ L.) Clearly such an embedding sends $\alpha \in$ L to one of its conjugates over K.
As a second question: How does an "embedding" of something that is already there $\alpha \in$ L $\subset \mathbb C$ get sent to a conjugate? Rather than call it a injective map or permutation?
Lastly in the proof of the theorem, he says: let $\sigma$ be an embedding of K in $\mathbb C$ ... and f be the monic irreducible polynomial for $\alpha$ over K. He then goes on to apply $\sigma$ to the coefficients of f to get g. But I thought (as above) that the embeddings fix K, so why would he say then say that g is irreducible (no problem) over $\sigma$K? Shouldn't that just be K?
As always thanks for your help and patience with what is probably a pretty obvious question.