On the Main Invariant of Elements Algebraic Over a Henselian Valued Field

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 L(Y ), where L is the residue field of the unique prolongation of v to a finite simple extension L of K and Y is transcendental over L (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 L[Y ] 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 L[Y ] with respect to w, is irreducible over K. Our results extend Eisenstein-Dumas and generalized Schönemann irreducibility criteria.

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“…The following already known lemma is proved in [1,Lemma 2.2]. For reader's convenience, we prove it here.…”

Section: Preliminary Resultsmentioning

confidence: 87%

On Liftings of Powers of Irreducible Polynomials

2013

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“…To see that any monic irreducible polynomial over K is a non-trivial lifting of some monic irreducible polynomial over the residue field of w, we use the notion of distinguished pairs which was introduced by Popescu and Zaharescu [21], for local fields in 1995 and was later generalized to arbitrary henselian valued fields (cf. [1] and [3]). Definition 1.7.…”

Section: Notations Definitions and Main Resultsmentioning

confidence: 99%

“…Later, this work was generalized by Aghigh et al (cf. [1]- [4]) to henselian valued fields. The notion of saturated distinguished chains is used to find results about irreducible polynomials and to obtain various invariants associated with elements of K. Recently Jakhar and Sangwan in [10], established a connection between distinguished pairs and key polynomials over a residually transcendental extension of v. In this paper, we associate distinguished pairs with abstract key polynomials over an extension w of v to K(X).…”

Section: Introductionmentioning

confidence: 99%

Abstract key polynomials and distinguished pairs

Mavi¹,

Bishnoi²

2021

Preprint

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“…Then by the strong triangle law, we have Lemma 3.2. Let g 0 w 0 0 g 1 w 1 1 g n w n n be a strict system of polynomial extensions over a henselian valued field K v and 1 be roots of g n g n−1 , respectively, such that v − 1 = K , with K defined by (1). If v g n > v g n 1 for some in K, then there exists a K-conjugate of such that v − > K Proof.…”

Section: Corollary Let Be As In Lemma 2d Thenmentioning

confidence: 99%

“…Popescu and Zaharescu [9], Ota [8], and Aghigh and Khanduja [1,2] have shown that one can associate several invariants to a defectless polynomial by means of complete distinguished chains defined below.…”

Section: Introductionmentioning

confidence: 99%

On Invariants and Strict Systems of Irreducible Polynomials Over Henselian Valued Fields

Khanduja

Khassa

2011

Communications in Algebra

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Let g x be a monic irreducible defectless polynomial over a henselian valued field K v , i.e., K is a defectless extension of K v for any root of g x . It is known that a complete distinguished chain for with respect to K v gives rise to several invariants associated with g x . Recently Ron Brown studied certain invariants of defectless polynomials by introducing strict systems of polynomial extensions. In this article, the authors establish a one-to-one correspondence between strict systems of polynomial extensions and conjugacy classes of complete distinguished chains. This correspondence leads to a simple interpretation of various results proved for strict systems. The authors give new characterizations of an invariant g introduced by Brown.

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On the Main Invariant of Elements Algebraic Over a Henselian Valued Field

Abstract: Let v be a henselian valuation of a field K with value group G, letv be the (unique)

Cited by 19 publications

References 9 publications

On Liftings of Powers of Irreducible Polynomials

On Liftings of Powers of Irreducible Polynomials

Abstract key polynomials and distinguished pairs

On Invariants and Strict Systems of Irreducible Polynomials Over Henselian Valued Fields

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