Separable Extension
Let
Let
be an embedding into an algebraic closure
Define
The separable degree of
Independence of the Embedding
Let
be two embeddings into algebraic closures
Claim
There exists a bijection
Construction of the Bijection
We have:
The map
is an isomorphism.
Since:
is an algebraic closure of , is an algebraic closure of ,
the isomorphism extends to an isomorphism
Thus we may replace
respectively.
Now define a map:
Verification
Let
We check that:
For
Thus
This gives a bijection.
Hence the separable degree is independent of the choice of embedding and algebraic closure.
Bounding the Number of Embeddings
Let:
be a field, an algebraic closure, algebraic over , .
Let
be the minimal polynomial of
If
and
is an extension of
must be a root of
Thus:
This gives the fundamental inequality:
Equality holds exactly when the extension is separable.