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 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.

CHAIN CONDITIONS ON NON-SUMMANDS

2011 ◽  
Vol 10 (03) ◽  
pp. 475-489 ◽  
Author(s):  
PINAR AYDOĞDU ◽  
A. ÇIĞDEM ÖZCAN ◽  
PATRICK F. SMITH
Keyword(s):  
Noetherian Ring ◽  
Jacobson Radical ◽  
Finitely Generated ◽  
Chain Conditions ◽  
Finite Dimensional

Let R be a ring. Modules satisfying ascending or descending chain conditions (respectively, acc and dcc) on non-summand submodules belongs to some particular classes [Formula: see text], such as the class of all R-modules, finitely generated, finite-dimensional and cyclic modules, are considered. It is proved that a module M satisfies acc (respectively, dcc) on non-summands if and only if M is semisimple or Noetherian (respectively, Artinian). Over a right Noetherian ring R, a right R-module M satisfies acc on finitely generated non-summands if and only if M satisfies acc on non-summands; a right R-module M satisfies dcc on finitely generated non-summands if and only if M is locally Artinian. Moreover, if a ring R satisfies dcc on cyclic non-summand right ideals, then R is a semiregular ring such that the Jacobson radical J is left t-nilpotent.

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.

scholarly journals Generating infinite monoids of cellular automata

2021 ◽  
pp. 2250215
Author(s):  
Alonso Castillo-Ramirez
Keyword(s):  
Cellular Automata ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Group Of Units ◽  
Finite Dimensional ◽  
Index Condition ◽  
Indicable Group

For a group [Formula: see text] and a set [Formula: see text], let [Formula: see text] be the monoid of all cellular automata over [Formula: see text], and let [Formula: see text] be its group of units. By establishing a characterization of surjunctive groups in terms of the monoid [Formula: see text], we prove that the rank of [Formula: see text] (i.e. the smallest cardinality of a generating set) is equal to the rank of [Formula: see text] plus the relative rank of [Formula: see text] in [Formula: see text], and that the latter is infinite when [Formula: see text] has an infinite decreasing chain of normal subgroups of finite index, condition which is satisfied, for example, for any infinite residually finite group. Moreover, when [Formula: see text] is a vector space over a field [Formula: see text], we study the monoid [Formula: see text] of all linear cellular automata over [Formula: see text] and its group of units [Formula: see text]. We show that if [Formula: see text] is an indicable group and [Formula: see text] is finite-dimensional, then [Formula: see text] is not finitely generated; however, for any finitely generated indicable group [Formula: see text], the group [Formula: see text] is finitely generated if and only if [Formula: see text] is finite.

scholarly journals Weyl modules and Weyl functors for hyper-map algebras

2020 ◽  
pp. 2140007
Author(s):  
Angelo Bianchi ◽  
Samuel Chamberlin
Keyword(s):  
Lie Algebra ◽  
Finitely Generated ◽  
Weyl Modules ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Universal Properties ◽  
Definition Of ◽  
Unitary Algebra

We investigate the representations of the hyperalgebras associated to the map algebras [Formula: see text], where [Formula: see text] is any finite-dimensional complex simple Lie algebra and [Formula: see text] is any associative commutative unitary algebra with a multiplicatively closed basis. We consider the natural definition of the local and global Weyl modules, and the Weyl functor for these algebras. Under certain conditions, we prove that these modules satisfy certain universal properties, and we also give conditions for the local or global Weyl modules to be finite-dimensional or finitely generated, respectively.

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.

Factors of Fields

1990 ◽  
Vol 33 (1) ◽  
pp. 79-83
Author(s):  
James K. Deveney ◽  
Joe Yanik
Keyword(s):  
Quotient Ring ◽  
Field Extension ◽  
Finitely Generated ◽  
Transcendental Extension ◽  
Rational Factor

AbstractLet L be a finitely generated extension of a field k. L is a k-rational factor if there is a field extension K of k such that the total quotient ring of L ꕕk K is a rational (pure transcendental) extension of K. We present examples of non-rational rational factors and explicitly determine both factors.

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.

Decision problems concerning S-arithmetic groups

1985 ◽  
Vol 50 (3) ◽  
pp. 743-772 ◽  
Author(s):  
Fritz Grunewald ◽  
Daniel Segal
Keyword(s):  
Additive Group ◽  
Nilpotent Groups ◽  
Isomorphism Problem ◽  
Algorithmic Problem ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
Finite Dimensional ◽  
Associative Rings ◽  
Finitely Presented ◽  
Torsion Free

This paper is a continuation of our previous work in [12]. The results, and some applications, have been described in the announcement [13]; it may be useful to discuss here, a little more fully, the nature and purpose of this work.We are concerned basically with three kinds of algorithmic problem: (1) isomorphism problems, (2) “orbit problems”, and (3) “effective generation”.(1) Isomorphism problems. Here we have a class of algebraic objects of some kind, and ask: is there a uniform algorithm for deciding whether two arbitrary members of are isomorphic? In most cases, the answer is no: no such algorithm exists. Indeed this has been one of the most notable applications of methods of mathematical logic in algebra (see [26, Chapter IV, §4] for the case where is the class of all finitely presented groups). It turns out, however, that when consists of objects which are in a certain sense “finite-dimensional”, then the isomorphism problem is indeed algorithmically soluble. We gave such algorithms in [12] for the following cases: = {finitely generated nilpotent groups}; = {(not necessarily associative) rings whose additive group is finitely generated}; = {finitely Z-generated modules over a fixed finitely generated ring}.Combining the methods of [12] with his own earlier work, Sarkisian has obtained analogous results with the integers replaced by the rationals: in [20] and [21] he solves the isomorphism problem for radicable torsion-free nilpotent groups of finite rank and for finite-dimensional Q-algebras.

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