Graded transcendental extensions of graded fields

Summary 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.

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Valued Differential Fields

Asymptotic Differential Algebra and Model Theory of Transseries ◽

10.23943/princeton/9780691175423.003.0007 ◽

2017 ◽

Author(s):

Matthias Aschenbrenner ◽

Lou van den Dries ◽

Joris van der Hoeven

Keyword(s):

Asymptotic Behavior ◽

Residue Field ◽

Field Extension ◽

Field Extensions ◽

Differential Field ◽

Differential Fields ◽

Dominant Part

This chapter deals with valued differential fields, starting the discussion with an overview of the asymptotic behavior of the function vsubscript P: Γ‎ → Γ‎ for homogeneous P ∈ K K{Y}superscript Not Equal To. The chapter then shows that the derivation of any valued differential field extension of K that is algebraic over K is also small. It also explains how differential field extensions of the residue field k give rise to valued differential field extensions of K with small derivation and the same value group. Finally, it discusses asymptotic couples, dominant part, the Equalizer Theorem, pseudocauchy sequences, and the construction of canonical immediate extensions.

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Ramification Groups of Abelian Local Field Extensions

Canadian Journal of Mathematics ◽

10.4153/cjm-1971-027-9 ◽

1971 ◽

Vol 23 (2) ◽

pp. 271-281 ◽

Cited By ~ 10

Author(s):

Murray A. Marshall

Keyword(s):

Prime Ideal ◽

Local Field ◽

Galois Extension ◽

Residue Class ◽

Abelian Extensions ◽

Ring Of Integers ◽

Field Extensions ◽

Valued Field ◽

Residue Class Field ◽

Image Position

Let k be a local field; that is, a complete discrete-valued field having a perfect residue class field. If L is a finite Galois extension of k then L is also a local field. Let G denote the Galois group GL|k. Then the nth ramification group Gn is defined bywhere OL, denotes the ring of integers of L, and PL is the prime ideal of OL. The ramification groups form a descending chain of invariant subgroups of G:1In this paper, an attempt is made to characterize (in terms of the arithmetic of k) the ramification filters (1) obtained from abelian extensions L\k.

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How Fields Can have a Product

Canadian Mathematical Bulletin ◽

10.4153/cmb-1978-045-7 ◽

1978 ◽

Vol 21 (2) ◽

pp. 253-254

Author(s):

Frank Zorzitto

Keyword(s):

Field Extension ◽

Universal Property ◽

Field Extensions

Let k be a field. Two field extensions E, F of k are said to have a product- in the category of field extensions of k (see e.g. [1, p. 30]) if and only if there exist a field extension P of k and two k -isomorphisms P→ E, P→ F satisfying the following universal property. For any field extension K of k and any pair of k-isomorphisms K→E, K→F, there exists a unique k-isomorphism K→P such that the diagrams below commute.

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Multivariable dimension polynomials and new invariants of differential field extensions

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171201006767 ◽

2001 ◽

Vol 27 (4) ◽

pp. 201-214 ◽

Cited By ~ 6

Author(s):

Alexander B. Levin

Keyword(s):

Field Extension ◽

Differential Polynomials ◽

Finitely Generated ◽

Field Extensions ◽

Differential Field ◽

Differential Dimension ◽

Characteristic Sets ◽

Differential Dimension Polynomial ◽

Dimension Polynomial

We introduce a special type of reduction in the ring of differential polynomials and develop the appropriate technique of characteristic sets that allows to generalize the classical Kolchin's theorem on differential dimension polynomial and find new differential birational invariants of a finitely generated differential field extension.

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Subfields and Invariants of Inseparable Field Extensions

Canadian Journal of Mathematics ◽

10.4153/cjm-1977-131-4 ◽

1977 ◽

Vol 29 (6) ◽

pp. 1304-1311 ◽

Cited By ~ 5

Author(s):

James K. Deveney ◽

John N. Mordeson

Keyword(s):

Field Extension ◽

Field Extensions

Let L/K be a field extension of characteristic p ≠ 0. The existence of intermediate fields over which L is regular, separable, or modular is important in recent Galois theories. For instance, see [1; 2; 3; 4; 7; 8; 9 and 14].

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Counting the number of distinct distances of elements in valued field extensions

Journal of Algebra ◽

10.1016/j.jalgebra.2018.05.010 ◽

2018 ◽

Vol 509 ◽

pp. 192-211 ◽

Cited By ~ 1

Author(s):

Anna Blaszczok ◽

Franz-Viktor Kuhlmann

Keyword(s):

Field Extensions ◽

Valued Field

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Algebraic Extensions

Formalized Mathematics ◽

10.2478/forma-2021-0004 ◽

2021 ◽

Vol 29 (1) ◽

pp. 39-47

Author(s):

Christoph Schwarzweller ◽

Agnieszka Rowińska-Schwarzweller

Keyword(s):

Field Theory ◽

Field Extension ◽

Field Extensions ◽

Intermediate Field ◽

Finite Set ◽

Algebraic Elements ◽

Roots Of A Polynomial

Summary In this article we further develop field theory in Mizar [1], [2], [3] towards splitting fields. We deal with algebraic extensions [4], [5]: a field extension E of a field F is algebraic, if every element of E is algebraic over F. We prove amongst others that finite extensions are algebraic and that field extensions generated by a finite set of algebraic elements are finite. From this immediately follows that field extensions generated by roots of a polynomial over F are both finite and algebraic. We also define the field of algebraic elements of E over F and show that this field is an intermediate field of E|F.

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On Roots of Polynomials over F[X]/ 〈p〉

Formalized Mathematics ◽

10.2478/forma-2019-0010 ◽

2019 ◽

Vol 27 (2) ◽

pp. 93-100

Author(s):

Christoph Schwarzweller

Keyword(s):

Irreducible Polynomial ◽

Field Extension ◽

Isomorphic Copy ◽

Field Extensions ◽

The Third ◽

Fourth Part ◽

Roots Of Polynomials ◽

Classical Proof ◽

Do So

Summary This is the first part of a four-article series containing a Mizar [3], [1], [2] 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 [9], [4], [6]. In this 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 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 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 i 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.

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