Extension of an order to a simple transcendental extension

1982 ◽  
pp. 113-118 ◽  
Author(s):  
Robert Gilmer
Keyword(s):  
Transcendental Extension ◽  
Simple Transcendental Extension

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.

scholarly journals A uniqueness problem in simple transcendental extensions of valued fields

1994 ◽  
Vol 37 (1) ◽  
pp. 13-23 ◽  
Author(s):  
Sudesh K. Khanduja
Keyword(s):  
Torsion Group ◽  
Affirmative Answer ◽  
Uniqueness Problem ◽  
Unique Extension ◽  
Valued Fields ◽  
Numerical Invariants ◽  
Transcendental Extension ◽  
Simple Transcendental Extension

Let υ0 be a valuation of a field K0 with value group G0 and υ be an extension of υ0 to a simple transcendental extension K0(x) having value group G such that G/G0 is not a torsion group. In this paper we investigate whether there exists t∈K0(x)/K0 with υ(t) non-torsion mod G0 such that υ is the unique extension to K0(x) of its restriction to the subfield K0(t). It is proved that the answer to this question is “yes” if υ0 is henselian or if υ0 is of rank 1 with G0 a cofinal subset of the value group of υ in the latter case, and that it is “no” in general. It is also shown that the affirmative answer to this problem is equivalent to a fundamental equality which relates some important numerical invariants of the extension (K, υ)/(K0, υ0).

On couplings on a simple transcendental extension

2003 ◽  
Vol 44 (3-4) ◽  
pp. 289-311 ◽  
Author(s):  
Christian Karpfinger
Keyword(s):  
Transcendental Extension ◽  
Simple Transcendental Extension

ON LIFTINGS OF POWERS OF IRREDUCIBLE POLYNOMIALS

2013 ◽  
Vol 12 (05) ◽  
pp. 1250222 ◽  
Author(s):  
ANUJ BISHNOI ◽  
SANJEEV KUMAR ◽  
SUDESH K. KHANDUJA
Keyword(s):  
Residue Field ◽  
Sufficient Conditions ◽  
Irreducible Polynomial ◽  
Simple Extension ◽  
Canonical Homomorphism ◽  
Irreducible Polynomials ◽  
Transcendental Extension ◽  
Simple Transcendental Extension ◽  
Irreducibility Criteria

Let v be a henselian valuation of arbitrary rank of a field K with valuation ring Rv having maximal ideal Mv. Using the canonical homomorphism from Rv onto Rv/Mv, one can lift any monic irreducible polynomial with coefficients in Rv/Mv to yield monic irreducible polynomials over Rv. Popescu and Zaharescu extended this approach and introduced the notion of lifting with respect to a residually transcendental prolongation w of v to a simple transcendental extension K(x) of K. As it is well known, the residue field of such a prolongation w is [Formula: see text], where [Formula: see text] is the residue field of the unique prolongation of v to a finite simple extension L of K and Y is transcendental over [Formula: see text] (see [V. Alexandru, N. Popescu and A. Zaharescu, A theorem of characterization of residual transcendental extension of a valuation, J. Math. Kyoto Univ.28 (1988) 579–592]). It is known that a lifting of an irreducible polynomial belonging to [Formula: see text] with respect to w, is irreducible over K. In this paper, we give some sufficient conditions to ensure that a given polynomial in K[x] satisfying these conditions which is a lifting of a power of some irreducible polynomial belonging to [Formula: see text] with respect to w, is irreducible over K. Our results extend Eisenstein–Dumas and generalized Schönemann irreducibility criteria.

scholarly journals On extensions of valuations to simple transcendental extensions

1989 ◽  
Vol 32 (1) ◽  
pp. 147-156 ◽  
Author(s):  
Sudesh K. Khanduja ◽  
Usha Garg
Keyword(s):  
Residue Field ◽  
Explicit Construction ◽  
Residue Theorem ◽  
Complete Field ◽  
Transcendental Extension ◽  
Simple Transcendental Extension

Let ν0 be a valuation of a field K0 with residue field k0 and value group Z, the group of rational integers. Let K0(x) be a simple transcendental extension of K0. In 1936, Maclane [3] gave a method to determine all real valuations V of K0(x) which are extensions of ν0. But his method does not seem to give an explicit construction of these valuations. In the present paper, assuming K0 to be a complete field with respect to ν0, we explicitly determine all extensions of ν0 to K0(x) which have Z as the value group and a simple transcendental extension of k0 as the residue field. If V is any extension of ν0 to K0(x) having Z as the value group and a transcendental extension of k0 as the residue field, then using the Ruled Residue theorem [4, 2, 5], we give a method which explicitly determines V on a subfield of K0(x) properly containing K0.

scholarly journals A Theorem on Valuation Rings and its Applications

1967 ◽  
Vol 29 ◽  
pp. 85-91 ◽  
Author(s):  
Masayoshi Nagata
Keyword(s):  
Valuation Ring ◽  
Algebraic Extension ◽  
Maximal Ideals ◽  
Valuation Rings ◽  
Transcendental Extension ◽  
Simple Transcendental Extension

In the present paper, we first prove the following THEOREM 1. Let K be a field, x a transcendental element over K, and V* a valuation ring of K(x). Set V= V* ∩ K. Denote by p* and p the maximal ideals of V* and V respectively. If (i) V*/p* is not algebraic over V/p and (ii) the value group of V* is isomorphic to Zn (Z=the module of rational integers), i.e., V* is of rank n and discrete in the generalized sense, then V*/p* is a simple transcendental extension of a finite algebraic extension of V/p.

scholarly journals On valuations of K(x)

1992 ◽  
Vol 35 (3) ◽  
pp. 419-426 ◽  
Author(s):  
Sudesh K. Khanduja
Keyword(s):  
Abelian Group ◽  
Finite Index ◽  
Residue Field ◽  
Torsion Group ◽  
Closed Field ◽  
Converse Problem ◽  
Valued Field ◽  
Transcendental Extension ◽  
Residue Fields ◽  
Simple Transcendental Extension

For a valued field (K, v), let Kv denote the residue field of v and Gv its value group. One way of extending a valuation v defined on a field K to a simple transcendental extension K(x) is to choose any α in K and any μ in a totally ordered Abelian group containing Gv, and define a valuation w on K[x] by w(Σici(x – α)i) = mini (v(ci) + iμ). Clearly either Gv is a subgroup of finite index in Gw = Gv + ℤμ or Gw/Gv is not a torsion group. It can be easily shown that K(x)w is a simple transcendental extension of Kv in the former case. Conversely it is well known that for an algebraically closed field K with a valuation v, if w is an extension of v to K(x) such that either K(x)w is not algebraic over Kv or Gw/Gv is not a torsion group, then w is of the type described above. The present paper deals with the converse problem for any field K. It determines explicitly all such valuations w together with their residue fields and value groups.

scholarly journals Generalized Hensel's lemma

1999 ◽  
Vol 42 (3) ◽  
pp. 469-480 ◽  
Author(s):  
Sudesh K. Khanduja ◽  
Jayanti Saha
Keyword(s):  
Valuation Ring ◽  
Analogous Result ◽  
Residue Field ◽  
Valued Field ◽  
Transcendental Extension ◽  
Simple Transcendental Extension ◽  
Hensel’S Lemma

Let (K, v) be a complete, rank-1 valued field with valuation ring Rv, and residue field kv. Let vx be the Gaussian extension of the valuation v to a simple transcendental extension K(x) defined by The classical Hensel's lemma asserts that if polynomials F(x), G0(x), H0(x) in Rv[x] are such that (i) vx(F(x) – G0(x)H0(x)) > 0, (ii) the leading coefficient of G0(x) has v-valuation zero, (iii) there are polynomials A(x), B(x) belonging to the valuation ring of vx satisfying vx(A(x)G0(x) + B(x)H0(x) – 1) > 0, then there exist G(x), H(x) in K[x] such that (a) F(x) = G(x)H(x), (b) deg G(x) = deg G0(x), (c) vx(G(x)–G0(x)) > 0, vx(H(x) – H0(x)) > 0. In this paper, our goal is to prove an analogous result when vx is replaced by any prolongation w of v to K(x), with the residue field of wa transcendental extension of kv.

Extending locally bounded field topologies to a simple transcendental extension

1981 ◽  
Vol 20 (5) ◽  
pp. 334-341
Author(s):  
V. I. Arnautov
Keyword(s):  
Transcendental Extension ◽  
Simple Transcendental Extension
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