§ Separable extension via embeddings into alg. closure
§ Defn by embeddings
- Let be a finite extension.
- It is separable iff a given embedding can be extended in ways (This number can be at most .)
- We call the numbe of ways to embed in via extending to be the separability degree of .
§ At most embeddings exist
- We will show for simple extensions that there are at most ways to extend into .
- We use two facts: first, is entirely determined by where it sends . Second, can only go to another root of its minimal polynomial . Thus, there are only finitely many choices, and the minimal polynomial has at most unique roots, and . Thus, there are at most choices of where can go to, which entirely determines . Thus there are at most choices for .
- Given a larger extension, write a sequence of extensions . Then, since and so on, can repeatedly apply the same argument to bound the number of choices of .
- In detail, for the case , consider the minimal polynomial of , . Then .
- Since fixes , and has coefficients from , we have that .
- Thus, in particular, .
- This implies that , or is a root of .
- Since , can only map to one of the other roots of .
- has at most unique roots [can have repeated roots, or some such, so could have fewer that that ].
- Further, is entirely determined by where it maps . Thus, there are at most ways to extend to .
§ Separability is transitive
- Given a tower , we fix an embedding . If both and are finite and separable, then extends into through in ways, and then again as in ways.
- This together means that we have ways to extend into , which is the maximum possible.
- Thus, is separable.
§ Separable by polynomial implies separable by embeddings
- Let every have minimal polynomial that is separable (ie, has distinct roots).
- Then we must show that allows us to extend any embedding in ways into
- Write as a tower of extensions. Let , and with .
- At each step, since the polynomial is separable, we have the maximal number of choices of where we send . Since degree is multiplicative, we have that .
- We build inductively as with .
- Then at step , which is has choices, since is separable over since its minimal polynomial is separable.
- This means that in toto, we have the correct number of choices for , which is what it means to be separable by embeddings.
§ Separable by embeddings implies separable by polynomial
- Let be separable in terms of embeddings. Consider some element , let its minimal polynomial be .
- Write . Since degree is multiplicative, we have .
- So given an embedding ,we must be able to extend it in ways.
- Since must send to a root of , and we need the total to be , we must have that has no repeated roots.
- If had repeated roots, then we will have fewer choices of thatn , which means the total count of choices for will be less than , thereby contradicting separability.
§ Finite extensions generated by separable elements are separable
- Let be separable, so there are ways to extend a map into .
- Since we have shown that separable by polyomial implies separable by embedding, we write . Each step is separable by the arguments given above in terms of counting automorphisms by where they send . Thus, the full is separable.