Soft Int-Field Extension

2021 ◽  
pp. 1-25
Author(s):  
Jayanta Ghosh ◽  
Dhananjoy Mandal ◽  
Tapas Kumar Samanta
Keyword(s):  
Field Extension ◽  
Field Extensions ◽  
Algebraic Element

The relation between soft element-wise field and soft int-field has been established and then some properties of soft int-field are studied. We define the notions of soft algebraic element and soft purely inseparable element of a soft int-field extension. Some characterizations of soft algebraic and soft purely inseparable int-field extensions are given. Lastly, we define soft separable algebraic int-field extension and study some of its properties.

scholarly journals On Monomorphisms and Subfields

2019 ◽  
Vol 27 (2) ◽  
pp. 133-137
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 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]/<p> 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]/<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 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]/<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.

Valued Differential Fields

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.

scholarly journals How Fields Can have a Product

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.

scholarly journals Multivariable dimension polynomials and new invariants of differential field extensions

2001 ◽  
Vol 27 (4) ◽  
pp. 201-214 ◽  
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.

Subfields and Invariants of Inseparable Field Extensions

1977 ◽  
Vol 29 (6) ◽  
pp. 1304-1311 ◽  
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].

scholarly journals Algebraic Extensions

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.

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

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]/<p> 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]/<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 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 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]/<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.

scholarly journals On the Intersection of Fields F with F [X]

2019 ◽  
Vol 27 (3) ◽  
pp. 223-228
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 third 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]/<p> 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]/<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.

The Order of Inseparability of Fields

1979 ◽  
Vol 31 (3) ◽  
pp. 655-662 ◽  
Author(s):  
James K. Deveney ◽  
John N. Mordeson
Keyword(s):  
Field Extension ◽  
Minimal Degree ◽  
Finitely Generated ◽  
Field Extensions ◽  
Finite Dimensional ◽  
Separable Extensions

Let L be a finitely generated field extension of a field K of characteristic p ≠ 0. By Zorn's Lemma there exist maximal separable extensions of K in L and L is finite dimensional purely inseparable over any such field. If ps is the smallest of the dimensions of L over such maximal separable extensions of K in L, then s is Wiel's order of inseparability of L/K [11]. Dieudonné [2] also investigated maximal separable extensions D of K in L and established that there must be at least one D such that L ⊆ Kp–∞(D) (such fields are termed distinguished). Kraft [5] showed that the distinguished maximal separable subfields are precisely those over which L is of minimal degree. This concept of distinguished subfield has been the basis of a number of results on the structure of inseparable field extensions, for example see [1], [3], [5], and [6].

scholarly journals Dissociation between red and white stimulus perception: A perimetric quantification of protanopic color vision deficiencies

PLoS ONE ◽  
2021 ◽  
Vol 16 (12) ◽  
pp. e0260362
Author(s):  
Denise Wetzel ◽  
Judith Ungewiss ◽  
Michael Wörner ◽  
Helmut Wilhelm ◽  
Ulrich Schiefer
Keyword(s):  
Visual Field ◽  
Color Vision ◽  
Field Extension ◽  
Control Group ◽  
Horizontal Meridian ◽  
Field Extensions ◽  
Kinetic Perimetry ◽  
Color Vision Deficiencies ◽  
The Right ◽  
Age Range

Significance Horizontal visual field extension was assessed for red and white stimuli in subjects with protanopia using semi-automated kinetic perimetry. In contrast to a conventional anomaloscope, the “red/white dissociation ratio” (RWR) allows to describe protanopia numerically. For the majority of subjects with protanopia a restriction for faint red stimuli was found. Purpose Comparing the horizontal visual field extensions for red and white stimuli in subjects with protanopia and those with normal trichromacy and assessing the related intra-subject intra-session repeatability. Methods The subjects were divided into groups with protanopia and with normal trichromacy, based on color vision testing (HMC anomaloscope, Oculus, Wetzlar/FRG). Two stimulus characteristics, III4e and III1e, according to the Goldmann-classification, were presented with semi-automated kinetic perimetry (Octopus 900 perimeter, Haag-Streit, Köniz/CH). They moved along the horizontal meridian, with an angular velocity of 3°/s towards the visual field center, starting from either the temporal or nasal periphery. If necessary, a 20° nasal fixation point offset was chosen to capture the temporal periphery of the visual field. For each condition the red/white dissociation ratio (RWR); Pat Appl. DPMA DRN 43200082D) between the extent of the isopter for red (RG610, Schott, Mainz/ FRG) and white stimuli along the horizontal meridian was determined. Results All data are listed as median/interquartile range: Five males with protanopia (age 22.1/4.5 years) and six males with normal trichromacy (control group, age 30.5/15.2 years) were enrolled. The RWR is listed for the right eye, as no clinically relevant difference between right and left eye occurred. Protanopes’ RWR for mark III4e (in brackets: control group) was 0.941/0.013 (0.977/0.019) and for mark III1e 0.496/0.062 (0.805/0.051), respectively. Conclusions In this exploratory “proof-of-concept study” red/white dissociation ratio perimetry is introduced as a novel technique aiming at assessing and quantifying the severity of protanopia. Further effort is needed to understand the magnitude of the observed red-/white dissociation and to extend this methodology to a wider age range of the sample and to anomalous trichromacies (protanomalia) with varying magnitude.

Sign in / Sign up

Export Citation Format

Share Document