Normal Extensions

Abstract

Summary In this article we continue the formalization of field theory in Mizar [1], [2], [4], [3]. We introduce normal extensions: an (algebraic) extension E of F is normal if every polynomial of F that has a root in E already splits in E . We proved characterizations (for finite extensions) by minimal polynomials [7], splitting fields, and fixing monomorphisms [6], [5]. This required extending results from [11] and [12], in particular that F [ T ] = { p ( a 1 , . . . a n ) | p ∈ F [ X ], a i ∈ T } and F ( T ) = F [ T ] for finite algebraic T ⊆ E . We also provided the counterexample that 𝒬(∛2) is not normal over 𝒬 (compare [13]).