Journal Details Format
Journal
eISSN
1898-9934
ISSN
1426-2630
First Published
09 Jun 2008
Publication timeframe
4 times per year
Languages
English Open Access

# On Monomorphisms and Subfields

###### Accepted: 27 May 2019
Journal Details Format
Journal
eISSN
1898-9934
ISSN
1426-2630
First Published
09 Jun 2008
Publication timeframe
4 times per year
Languages
English

This is the second 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 pF [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 pF [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 ⊆ [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 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]/<p> and therefore consider F as a subfield of F [X]/<p>”. Interestingly, to do so we need to assume that FE = ∅, in particular Kronecker’s construction can be formalized for fields F with FF [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 FE 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 pF [X]\F has a root.

#### MSC 2010

 Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.10.1007/s10817-017-9440-6

 Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.10.1007/s10817-015-9345-1

 Nathan Jacobson. Basic Algebra I. Dover Books on Mathematics, 1985.Search in Google Scholar

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• #### Algorithm NextFit for the Bin Packing Problem

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