Separating p-Bases and Transcendental Extension Fields

1972 ◽  
Vol 31 (2) ◽  
pp. 417
Author(s):  
J. N. Mordeson ◽  
B. Vinograde
Keyword(s):  
Transcendental Extension

Decidable fragments of field theories

1990 ◽  
Vol 55 (3) ◽  
pp. 1007-1018 ◽  
Author(s):  
Shih-Ping Tung
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Field Theories ◽  
Extension Field ◽  
Radical Extension ◽  
Transcendental Extension

AbstractWe say φ is an ∀∃ sentence if and only if φ is logically equivalent to a sentence of the form ∀x∃yψ(x, y), where ψ(x, y) is a quantifier-free formula containing no variables except x and y. In this paper we show that there are algorithms to decide whether or not a given ∀∃ sentence is true in (1) an algebraic number field K, (2) a purely transcendental extension of an algebraic number field K, (3) every field with characteristic 0, (4) every algebraic number field, (5) every cyclic (abelian, radical) extension field over Q, and (6) every field.

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 IRREDUCIBLE FACTORS OF POLYNOMIALS OVER COMPLETE FIELDS

2012 ◽  
Vol 12 (01) ◽  
pp. 1250125 ◽  
Author(s):  
SUDESH K. KHANDUJA ◽  
SANJEEV KUMAR
Keyword(s):  
Algebraic Number ◽  
Number Fields ◽  
Manuscripta Math ◽  
Algebraic Number Fields ◽  
Valued Fields ◽  
Valued Field ◽  
Transcendental Extension ◽  
Simple Transcendental Extension ◽  
Irreducibility Criterion ◽  
Hensel’S Lemma

Let (K, v) be a complete rank-1 valued field. In this paper, we extend classical Hensel's Lemma to residually transcendental prolongations of v to a simple transcendental extension K(x) and apply it to prove a generalization of Dedekind's theorem regarding splitting of primes in algebraic number fields. We also deduce an irreducibility criterion for polynomials over rank-1 valued fields which extends already known generalizations of Schönemann Irreducibility Criterion for such fields. A refinement of Generalized Akira criterion proved in Khanduja and Khassa [Manuscripta Math.134(1–2) (2010) 215–224] is also obtained as a corollary of the main result.

An abelian subextension of the dyadic division field of a hyperelliptic Jacobian

2019 ◽  
Vol 69 (2) ◽  
pp. 357-370
Author(s):  
Jeffrey Yelton
Keyword(s):  
Hyperelliptic Curve ◽  
Symmetric Functions ◽  
Explicit Description ◽  
Transcendental Extension ◽  
The Way

Abstract Given a field k of characteristic different from 2 and an integer d ≥ 3, let J be the Jacobian of the “generic” hyperelliptic curve given by $\begin{array}{} \displaystyle y^2 = \prod_{i = 1}^d (x - \alpha_i) \end{array}$ , where the αi’s are transcendental and independent over k; it is defined over the transcendental extension K/k generated by the symmetric functions of the αi’s. We investigate certain subfields of the field K∞ obtained by adjoining all points of 2-power order of J(K̄). In particular, we explicitly describe the maximal abelian subextension of K∞ / K(J[2]) and show that it is contained in K(J[8]) (resp. K(J[16])) if g ≥ 2 (resp. if g = 1). On the way we obtain an explicit description of the abelian subextension K(J[4]), and we describe the action of a particular automorphism in Gal(K∞ / K) on these subfields.

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 The invariants of orthogonal group actions

1993 ◽  
Vol 48 (2) ◽  
pp. 313-319 ◽  
Author(s):  
Li Chiang ◽  
Yu-Ching Hung
Keyword(s):  
Quadratic Form ◽  
Finite Field ◽  
Rational Function ◽  
Linear Group ◽  
Function Field ◽  
Orthogonal Group ◽  
Group Actions ◽  
Rational Function Field ◽  
Transcendental Extension

Let Fq be the finite field of order q, an odd number, Q a non-degenerate quadratic form on , O(n, Q) the orthogonal group defined by Q. Regard O(n, Q) as a linear group of Fq -automorphisms acting linearly on the rational function field Fq(x1, …, xn). We shall prove that the invariant subfield Fq(x1,…, xn)O(n, Q) is a purely transcendental extension over Fq for n = 5 by giving a set of generators for it.

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