On the intersection of fields F with F[X]
AbstractThis is the third part of a four-article series containing a Mizar , ,  formalization of Kronecker’s construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F[X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker’s classical proof using F[X]/< p > as the desired field extension E , , . In the first part we show that an irreducible polynomial p ∈ F[X]\F has a root over F[X]/< p >. Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ F[X]/< p > as sets, so F is not a subfield of F[X]/< p >, and hence formally p is not even a polynomial over F[X]/< p >. Consequently, we translate p along the canonical monomorphism φ : F → F[X]/< p > and show that the translated polynomial φ(p) has a root over F[X]/< p >. Because F is not a subfield of F[X]/< p > we construct in the second part the field (E\φF)∪F for a given monomorphism φ : F −→ E and show that this field both is isomorphic to F and includes F as a subfield. In the literature this part of the proof usually consists of saying that “one can identify F with its image φF in F[X]/< p > and therefore consider F as a subfield of F[X]/< p > ”. Interestingly, to do so we need to assume that F ∩ E = ∅, in particular Kronecker’s construction can be formalized for fields F with F ∩ F[X] = ∅. Surprisingly, as we show in this third part, this condition is not automatically true for arbitrary fields F: With the exception of Z2 we construct for every field F an isomorphic copy F' of F with F' ∩ F'[X] ≠ ∅. We also prove that for Mizar’s representations of Zn, Q and R we have Zn ∩ Zn[X] = ∅, Q ∩ Q[X] = ∅ and R ∩ R[X] = ∅, respectively. In the fourth part we finally define field extensions: E is a field extension of F iff F is a subfield of E. Note, that in this case we have F ⊆ E as sets, and thus a polynomial p over F is also a polynomial over E. We then apply the construction of the second part to F[X]/< p > with the canonical monomorphism φ : F → F[X]/< p >. Together with the first part this gives – for fields F with F ∩ F[X] = ∅ – a field extension E of F in which p ∈ F[X]\F has a root.
|Journal series||Formalized Mathematics, ISSN 1426-2630, e-ISSN 1898-9934, (N/A 20 pkt, Not active)|
|Publication size in sheets||0.5|
|Keywords in English||roots of polynomials, field extensions, Kronecker’s construction|
|License||Journal (articles only); published final; ; with publication|
|Score||= 20.0, 04-03-2020, ArticleFromJournal|
|Publication indicators||: 2018 = 0.169|
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