Splitting Fields

Abstract

In this article we further develop field theory in Mizar [1], [2]: we prove existence and uniqueness of splitting fields. We define the splitting field of a polynomial p ∈ F[X] as the smallest field extension of F, in which p splits into linear factors. From this follows, that for a splitting field E of p we have E = F(A) where A is the set of p’s roots. Splitting fields are unique, however, only up to isomorphisms; to be more precise up to F-isomorphims i.e. isomorphisms i with i|F = IdF . We prove that two splitting fields of p ∈ F[X] are F-isomorphic using the well-known technique [4], [3] of extending isomorphisms from F1 −→ F2 to F1(a) −→ F2(b) for a and b being algebraic over F1 and F2, respectively.