<abstract xmlns="http://www.w3.org/1999/xhtml">This is the fourth part of a four-article series containing a Mizar [3], [2], [1] formalization of Kronecker’s construction about roots of polynomials in field extensions, i.e. that for every field <italic>F</italic> and every polynomial <italic>p</italic> ∈ <italic>F</italic> [<italic>X</italic>]<italic>\F</italic> there exists a field extension <italic>E</italic> of <italic>F</italic> such that <italic>p</italic> has a root over <italic>E</italic>. The formalization follows Kronecker’s classical proof using <italic>F</italic> [<italic>X</italic>]<italic>/&lt;p></italic> as the desired field extension <italic>E</italic> [6], [4], [5].In the first part we show that an irreducible polynomial <italic>p</italic> ∈ <italic>F</italic> [<italic>X</italic>]<italic>\F</italic> has a root over <italic>F</italic> [<italic>X</italic>]<italic>/&lt;p></italic>. Note, however, that this statement cannot be true in a rigid formal sense: We do not have <italic>F</italic> ⊆ <italic>F</italic> [<italic>X</italic>]<italic>/ &lt; p ></italic> as sets, so <italic>F</italic> is not a subfield of <italic>F</italic> [<italic>X</italic>]<italic>/&lt;p></italic>, and hence formally <italic>p</italic> is not even a polynomial over <italic>F</italic> [<italic>X</italic>]<italic>/ &lt; p ></italic>. Consequently, we translate <italic>p</italic> along the canonical monomorphism <italic>ϕ</italic>: <italic>F → F</italic> [<italic>X</italic>]<italic>/&lt;p></italic> and show that the translated polynomial <italic>ϕ</italic> (<italic>p</italic>) has a root over <italic>F</italic> [<italic>X</italic>]<italic>/&lt;p></italic>.Because <italic>F</italic> is not a subfield of <italic>F</italic> [<italic>X</italic>]<italic>/&lt;p></italic> we construct in the second part the field (<italic>E \ ϕF</italic>)∪<italic>F</italic> for a given monomorphism <italic>ϕ</italic>: <italic>F → E</italic> and show that this field both is isomorphic to <italic>F</italic> and includes <italic>F</italic> as a subfield. In the literature this part of the proof usually consists of saying that “one can identify <italic>F</italic> with its image <italic>ϕF</italic> in <italic>F</italic> [<italic>X</italic>]<italic>/&lt;p></italic> and therefore consider <italic>F</italic> as a subfield of <italic>F</italic> [<italic>X</italic>]<italic>/&lt;p></italic>”. Interestingly, to do so we need to assume that <italic>F</italic> ∩ <italic>E</italic> = ∅, in particular Kronecker’s construction can be formalized for fields <italic>F</italic> with <italic>F ∩ F</italic> [<italic>X</italic>] = ∅.Surprisingly, as we show in the third part, this condition is not automatically true for arbitrary fields <italic>F</italic> : With the exception of ℤ2 we construct for every field <italic>F</italic> an isomorphic copy <italic>F′</italic> of <italic>F</italic> with <italic>F′ ∩ F′</italic> [<italic>X</italic>] ≠ ∅. We also prove that for Mizar’s representations of ℤ<italic>n</italic>, ℚ and ℝ we have ℤ<italic>n ∩</italic> ℤ<italic>n</italic>[<italic>X</italic>] = ∅, ℚ <italic>∩</italic> ℚ[<italic>X</italic>] = ∅ and ℝ <italic>∩</italic> ℝ[<italic>X</italic>] = ∅, respectively.In this fourth part we finally define field extensions: <italic>E</italic> is a field extension of <italic>F</italic> iff <italic>F</italic> is a subfield of <italic>E</italic>. Note, that in this case we have <italic>F</italic> ⊆ <italic>E</italic> as sets, and thus a polynomial <italic>p</italic> over <italic>F</italic> is also a polynomial over <italic>E</italic>. We then apply the construction of the second part to <italic>F</italic> [<italic>X</italic>]<italic>/&lt;p></italic> with the canonical monomorphism <italic>ϕ</italic>: <italic>F → F</italic> [<italic>X</italic>]<italic>/&lt;p></italic>. Together with the first part this gives – for fields <italic>F</italic> with <italic>F ∩ F</italic> [<italic>X</italic>] = ∅ – a field extension <italic>E</italic> of <italic>F</italic> in which <italic>p</italic> ∈ <italic>F</italic> [<italic>X</italic>]<italic>\F</italic> has a root.</abstract>

This is the fourth part of a four-article series containing a Mizar [3], [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 [6], [4], [5].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 ⊆ 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 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]/ 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 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 this 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.

Field Extensions and Kronecker’s Construction

Formalized Mathematics

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License.

{"article-title":"Field Extensions and Kronecker\u2019s Construction"}

This is the fourth part of a four-article series containing a Mizar [3], [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 [6], [4], [5].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 ⊆ 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 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]/ 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 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 this 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.

This is the fourth part of a four-article series containing a Mizar [3], [2], [1] formalization of Kronecker’s construction about roots of polynomials in field...

Field Extensions and Kronecker’s Construction

Published Online: Feb 17, 2020

Page range: 229 - 235

Accepted: Aug 29, 2019

DOI: https://doi.org/10.2478/forma-2019-0022

Keywords
roots of polynomials, field extensions, Kronecker’s construction

© 2019 Christoph Schwarzweller, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License.

Field Extensions and Kronecker’s Construction

Published Online: Feb 17, 2020

Page range: 229 - 235

Accepted: Aug 29, 2019

DOI: https://doi.org/10.2478/forma-2019-0022

Keywordsroots of polynomials, field extensions, Kronecker’s construction

© 2019 Christoph Schwarzweller, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License.

Keywords
roots of polynomials, field extensions, Kronecker’s construction