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http://hdl.handle.net/11320/8132
Tytuł: | On Monomorphisms and Subfields |
Autorzy: | Schwarzweller, Christoph |
Słowa kluczowe: | roots of polynomials field extensions Kronecker’s construction 12E05 12F05 68T99 03B35 |
Data wydania: | 2019 |
Data dodania: | 29-lip-2019 |
Wydawca: | DeGruyter Open |
Źródło: | Formalized Mathematics, Volume 27, Issue 2, Pages 133 - 137 |
Abstrakt: | This is the second part of a four-article series containing a Mizar [2], [1] 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]/ as the desired field extension E [5], [3], [4].In the first part we show that an irreducible polynomial p ∈ F [X]\F has a root over F [X]/. Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ [X]/ as sets, so F is not a subfield of F [X]/, and hence formally p is not even a polynomial over F [X]/. Consequently, we translate p along the canonical monomorphism ϕ : F → F [X]/ and show that the translated polynomial ϕ (p) has a root over F [X]/.Because F is not a subfield of F [X]/ we construct in this 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]/ and therefore consider F as a subfield of F [X]/”. 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 the third part, this condition is not automatically true for arbitray fields F : With the exception of 2 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 n, and we have n ∩ n[X] = ∅, ∩ [X] = ∅ and ∩ [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]/ with the canonical monomorphism ϕ : F → F [X]/. 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. |
Afiliacja: | Institute of Informatics, University of Gdańsk, Poland |
URI: | http://hdl.handle.net/11320/8132 |
DOI: | 10.2478/forma-2019-0014 |
ISSN: | 1426-2630 |
e-ISSN: | 1898-9934 |
metadata.dc.identifier.orcid: | 0000-0001-9587-8737 |
Typ Dokumentu: | Article |
Występuje w kolekcji(ach): | Formalized Mathematics, 2019, Volume 27, Issue 2 |
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