Separable Extensions, Separable Subfield
Separable Subfield
Let u and v be separable elements over K.
Let F be the normal closure of K(u,v).
If u and v are separable then F is the splitting field for a set of separable polynomials,
and is galois.
A galois extension is separable, so everything in F is separable.
This includes u+v u-v u*v u/v.
Let F/K be any field extension.
The set of elements E in F that are separable over K are closed under addition and multiplication,
and form an intermediate field extension.
This is the separable subfield of F.