Normal Extensions

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(a₁, . . . 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]).