A well‐known result of Ehrenfeucht states that a difference polynomial f(X)‐g(Y) in two variables X, Y with complex coefficients is irreducible if the degrees of f and g are coprime. Panaitopol and Stefǎnescu generalized this result, by giving an irreducibility condition for a larger class of polynomials called “generalized difference polynomials”. This paper gives an irreducibility criterion for more general polynomials, of which the criterion of Panaitopol and Stefǎnescu is a special case.

Let v be a henselian valuation of a field K, and [Formula: see text] be the (unique) extension of v to a fixed algebraic closure [Formula: see text] of K. For an element [Formula: see text], a chain [Formula: see text] of elements of [Formula: see text] such that θi is of minimal degree over K with the property that [Formula: see text] and θm ∈ K, is called a complete distinguished chain for θ with respect to (K, v). In 1995, Popescu and Zaharescu proved the existence of a complete distinguished chain for each [Formula: see text] when (K, v) is a complete discrete rank one valued field (cf. [10]). In this paper, for a henselian valued field (K, v) of arbitrary rank, we characterize those elements [Formula: see text] for which there exists a complete distinguished chain. It is shown that a complete distinguished chain for θ gives rise to several invariants associated to θ which are same for all the K-conjugates of θ.

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Let [Formula: see text] denote the ring of algebraic integers of an algebraic number field [Formula: see text], where [Formula: see text] is a root of an irreducible trinomial [Formula: see text] belonging to [Formula: see text]. In this paper, we give necessary and sufficient conditions involving only [Formula: see text] for a given prime [Formula: see text] to divide the index of the subgroup [Formula: see text] in [Formula: see text]. In particular, we deduce necessary and sufficient conditions for [Formula: see text] to be equal to [Formula: see text].

a b s t r a c tLet v be a real valuation of a field K with valuation ring R v . Let K (θ ) be a finite separable extension of K with θ integral over R v and F (x) be the minimal polynomial of θ over K . Using Newton polygons and residually transcendental prolongations of v to a simple transcendental extension K (x) of K together with liftings with respect to such prolongations, we describe a method to determine all prolongations of v to K (θ ) along with their residual degrees and ramification indices over v. The problem is classical but our approach uses new ideas. The paper gives an analogue of Ore's Theorem when the base field is an arbitrary rank-1 valued field and extends the main result of [S.D. Cohen, A. Movahhedi, A. Salinier, Factorization over local fields and the irreducibility of generalized difference polynomials, Mathematika 47 (2000) .

Let [Formula: see text] be an algebraic number field with [Formula: see text] in the ring [Formula: see text] of algebraic integers of [Formula: see text] and [Formula: see text] be the minimal polynomial of [Formula: see text] over the field [Formula: see text] of rational numbers. In 1977, Uchida proved that [Formula: see text] if and only if [Formula: see text] does not belong to [Formula: see text] for any maximal ideal [Formula: see text] of the polynomial ring [Formula: see text] (see [Osaka J. Math. 14 (1977) 155–157]). In this paper, we apply the above result to obtain some necessary and sufficient conditions involving the coefficients of [Formula: see text] for [Formula: see text] to equal [Formula: see text] when [Formula: see text] is a trinomial of the type [Formula: see text]. In the particular case when [Formula: see text], it is deduced that [Formula: see text] is an integral basis of [Formula: see text] if and only if either (i) [Formula: see text] and [Formula: see text] or (ii) [Formula: see text] divides [Formula: see text] and [Formula: see text].

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