Inseparable splitting theory

Abstract:

If L is a purely inseparable field extension of K, we show that, for large enough extensions E of K, the E algebra $L{ \otimes _K}E$ splits to become a truncated polynomial algebra. In fact, there is a unique smallest extension E of K which splits $L/K$ and we call this the splitting field $S(L/K)$ of $L/K$. Now $L \subseteq S(L/K)$ and the extension $S(L/K)$ of K is also purely inseparable. This allows us to repeat the splitting field construction and obtain inductively a tower of fields. We show that the tower stabilizes in a finite number of steps and we study questions such as how soon must the tower stabilize. We also characterize in many ways the case when L is its own splitting field. Finally, we classify all K algebras A which split in a similar way to purely inseparable field extensions.