/sci/ - Science & Math

Let's say I have a field K and its extension L/K. The field K and the extended field L are both of characteristic 0. Simple, nice infinite fields that we love.

I have proved that any extension of K of degree two is automatically normal.

Now, the question is "is this in general true for any extension of degree greater than 2?".

Answer is obviously no. But how do I prove it?

My thoughts. For example let's say we extended K by an element and now we have basis 1, a, a^2. Degree is three.

Basically, normality of an extension is the fact that we can create automorphisms from one roots\ to other roots. Only in such case we can swap them if the polynomial is irreducible. So if we have a root in our extension field L over K of degree 3, a nice automorphism (conjugation) that we used in degree 2 case is no longer available. we kinda missing something, that must extend our L into new field so we can unlock these automorphisms.

Now the question, how to mathematically show this "missingness" ?