Totally Inseparable Extensions

(Section 52) Totally Inseparable Extensions

(Def 52.1) A finite extension E of a field F is a totally inseparable extension of F if {E : F} = 1 < [E : F] . an element \alpha of \bar{F} is totally inseparable over F if F(\alpha) is totally inseparable over F.

(Thm 52.3) If K is a finite extension of E, E is a finite extension of F, and F < E < K, then K is totally inseparable over F iff K is totally inseparable over E and E is totally inseparable over F.

(Cor 52.4) If E is a finite extension of F, then E is totally inseparable over F iff each \alpha in E, \alpha \neq F, is totally inseparable over F.

(subsection) Separable Closures

(Thm 52.5) Let F have characteristic p \neq 0, and let E be a finite extension of F. Then \alpha \in E, \alpha \notin F, is totally inseparable over F iff there is sime integer t \ge 1 s.t. \alpha^{p^{i}} \in F.

Furthermore, there is a unique extension K of F, with F \le K \le E, s.t. K is separable over F, and either E = K or E is totally inseparable over K.

(Def 52.6) The unique field K of (Thm 52.5) is the separable closure of F in E.