Representation Matters: An Unexpected Property of Polynomial Rings and its Consequences for Formalizing Abstract Field Theory

In this paper we develop a Mizar formalization of Kronecker's construction, which states that for every field F and irreducible polynomial p ∈ F[X] there exists a field extension E of F such that p has a root over E. It turns out that to prove the correctness of the construction the field F needs to provide a disjointness condition, namely F ∩F[X] = Ø. Surprisingly this property does not hold for arbitrary representations of a field F: We construct for almost every field F another representation F
^'
, i.e. an isomorphic copy F
^'
of F, not satisfying this condition. As a consequence to F
^'
our formalization of Kronecker's construction cannot be applied. All proofs have been carried out in the Mizar system. Based on Mizar's representation of the fields Z
_p
, Q and R we also have proven that Z
_p
∩ Z
_p
[X] = Ø, Q ∩ Q[X] = Ø, and R ∩ R[X] = Ø respectively.