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 On residually transcendental valued function fields of conics

1996 ◽  
Vol 38 (2) ◽  
pp. 137-145
Author(s):  
Sudesh K. Khanduja
Keyword(s):  
Valuation Ring ◽  
Function Fields ◽  
Residue Field ◽  
Field Extension ◽  
Transcendence Degree ◽  
Finitely Generated ◽  
Uniqueness Property ◽  
Transcendental Extension

Let K/Kobe a finitely generated field extension of transcendence degree 1. Let u0 be a valuation of Koand v a valuation of Kextending v0such that the residue field of vis a transcendental extension ofthe residue field k0of vo/such a prolongation vwill be called a residually transcendental prolongation of v0. Byan element with the uniqueness propertyfor (K, v)/(K0, v0) (or more briefly for v/v0)we mean an element / of Khaving u-valuation 0 which satisfies (i) the image of tunder the canonicalhomomorphism from the valuation ring of vonto the residue field of v(henceforth referred to as the v-residue ot t) is transcendental over ko; that is vcoincides with the Gaussian valuation on the subfield K0(t) defined by (ii) vis the only valuation of K (up to equivalence) extending the valuation .

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.

Hilbert rings and G(oldman)-rings issued from amalgamated algebras

2018 ◽  
Vol 17 (02) ◽  
pp. 1850023 ◽  
Author(s):  
L. Izelgue ◽  
O. Ouzzaouit
Keyword(s):  
New York ◽  
Commutative Algebra ◽  
Quotient Ring ◽  
Sufficient Conditions ◽  
Ring Homomorphism ◽  
Necessary And Sufficient Conditions ◽  
Local Rings ◽  
Finitely Generated ◽  
Necessary And Sufficient ◽  
The Way

Let [Formula: see text] and [Formula: see text] be two rings, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] be a ring homomorphism. The ring [Formula: see text] is called the amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text]. It was proposed by D’anna and Fontana [Amalgamated algebras along an ideal, Commutative Algebra and Applications (W. de Gruyter Publisher, Berlin, 2009), pp. 155–172], as an extension for the Nagata’s idealization, which was originally introduced in [Nagata, Local Rings (Interscience, New York, 1962)]. In this paper, we establish necessary and sufficient conditions under which [Formula: see text], and some related constructions, is either a Hilbert ring, a [Formula: see text]-domain or a [Formula: see text]-ring in the sense of Adams [Rings with a finitely generated total quotient ring, Canad. Math. Bull. 17(1) (1974)]. By the way, we investigate the transfer of the [Formula: see text]-property among pairs of domains sharing an ideal. Our results provide original illustrating examples.

Rings with A Finitely Generated Total Quotient Ring

1974 ◽  
Vol 17 (1) ◽  
pp. 1-4 ◽  
Author(s):  
John Conway Adams
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Quotient Ring ◽  
Finitely Generated ◽  
Zero Divisors ◽  
Rank One ◽  
Definition Of

Let R be a commutative ring with non-zero identity and let K be the total quotient ring of R. We call R a G-ring if K is finitely generated as a ring over R. This generalizes Kaplansky′s definition of G-domain [5].Let Z(R) be the set of zero divisors in R. Following [7] elements of R—Z(R) and ideals of R containing at least one such element are called regular. Artin-Tate's characterization of Noetherian G-domains [1, Theorem 4] carries over with a slight adjustment to characterize a Noetherian G-ring as being semi-local in which every regular prime ideal has rank one.

scholarly journals On derivations of fields of algebraic functions

1980 ◽  
Vol 23 (3) ◽  
pp. 239-241
Author(s):  
Julio R. Bastida
Keyword(s):  
Field Extension ◽  
Restriction Mapping ◽  
Finitely Generated ◽  
Algebraic Functions ◽  
Transcendency Degree

In this note, we shall prove that the transcendency degree of a finitely generated field extension is equal to a certain integer associated with a restriction mapping of spaces of derivations.

FAILURE OF CANCELLATION FOR QUARTIC AND HIGHER-DEGREE ORDERS

2002 ◽  
Vol 01 (04) ◽  
pp. 469-481 ◽  
Author(s):  
RYAN KARR
Keyword(s):  
Mild Condition ◽  
Residue Field ◽  
Unit Group ◽  
Field Extension ◽  
Integral Closure ◽  
Principal Ideal Domain ◽  
Finitely Generated ◽  
Quotient Field ◽  
Almost All ◽  
Finitely Generated Modules

Let D be a principal ideal domain with quotient field F and suppose every residue field of D is finite. Let K be a finite separable field extension of F of degree at least 4 and let [Formula: see text] denote the integral closure of D in K. Let [Formula: see text] where f ∈ D is a nonzero nonunit. In this paper we show, assuming a mild condition on f, that cancellation of finitely generated modules fails for R, that is, there exist finitely generated R-modules L, M, and N such that L ⊕ M ≅ L ⊕ N and yet M ≇ N. In case the unit group of D is finite, we show that cancellation fails for almost all rings of the form [Formula: see text], where p ∈ D is prime.

scholarly journals On Ramification Theory in Projective Orders, II

1972 ◽  
Vol 46 ◽  
pp. 147-153
Author(s):  
Shizuo Endo
Keyword(s):  
Commutative Ring ◽  
Quotient Ring ◽  
Finitely Generated ◽  
Ramification Theory

Let R be a commutative ring and K be the total quotient ring of R. Let Σ be a separable K-algebra which is a finitely generated projective, faithful K-module and Λ be an R-order in DΛ/R. We denote by DΛ/R the Dedekind different of Λ and by NΛ/R the Noetherian different of Λ.

The u-invariant of p-adic function fields

2013 ◽  
Vol 2013 (679) ◽  
pp. 65-73 ◽  
Author(s):  
David B. Leep
Keyword(s):  
Quadratic Form ◽  
Recent Result ◽  
Quadratic Forms ◽  
Function Fields ◽  
Field Extension ◽  
Finitely Generated ◽  
Nontrivial Zero

Abstract Over a finitely generated field extension in m variables over a p-adic field, any quadratic form in more than 2m + 2 variables has a nontrivial zero. This bound is sharp. We extend this result to a wider class of fields. A key ingredient to our proofs is a recent result of Heath-Brown on systems of quadratic forms over p-adic fields.

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].

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