So now we are ready to give the main definition of this course.
if and only if L is Galois.
This is a very easy theorem.
on the set of homomorphisms.
Of K-homomorphisms from L to K-bar.
of an orbit of such action.
than the cardinality of the set itself.
holds whenever this action is transitive.
that L is normal over K.
is less or equal than the number of homomorphisms.
And this is less or equal than the degree of the extension.
then it's less or equal.
to an automorphism of L.
Well, to see this, we embed L into an algebraic closure K-bar.
but all those maps have the same image, namely L, right?
extends to an automorphism of L.
then x is a purely unseparable element.
its minimal polynomial must have x as the only root.
has a single root x.
of L over K is just K itself.
Well, maybe you have not seen this notation.
then X^G is the set of invariants.
those x which are not moved by G.
automorphisms of L over K.
restate what I just have written here.