A field is said to be perfect if it has no non-separable algebraic extensions. Aside from being very useful as ground fields, they have a nice classification: they are exactly the fields of characteristic $0$ and the fields of characteristic $p>0$ that are closed under $p^{th}$ roots.
Along these lines I am wondering if there is a nice characterization of fields that are 'normally perfect' - fields with no non-normal algebraic extensions.
Just as a field being perfect is equivalent to it having no non-separable finite extensions, a field being 'normally perfect' is equivalent to it having no non-normal finite extensions, so it is easy to see that all finite fields are 'normally perfect'. It is also a trivial fact that all algebraically closed fields and all real closed fields are 'normally perfect'. But beyond this I'm having a difficult time seeing what fields could be 'normally perfect' or finding nice necessary conditions for these fields.