QUESTION: Suppose K/F is a non-trivial finite degree purely inseparable extension. Prove that there is a purely inseparable degree p extension of K.
I know, or can prove, that [K:F] is a power of p as well as the standard equivalences that usually come when defining purely inseparable extensions. However, this problem is stumping me. Can someone help me see why this purely inseparable extension being non-trivial and finite degree means K is not algebraically closed? And furthermore why there is an extension of K of degree p?