In this paper, further insight is obtained into the earlier approach of studying residually transcendental extensions of a valuation v of a field K to a simple transcendental extension K(x) of K by means of minimal pairs, thereby introducing new invariants corresponding to any element of an algebraic closure of K. It is also shown that these invariants are of independent interest as well. A characterization of those elements a belonging to is given such that there exists a minimal pair (a, δ) for some δ in the divisible closure of the value group of v.

Let v be a Henselian valuation of any rank of a field K and its unique prolongation to a fixed algebraic closure of K having value group . For any subfield L of , let R(L) denote the residue field of the valuation obtained by restricting to L. Using the canonical homomorphism from the valuation ring of v onto its residue field R(K), one can lift any monic irreducible polynomial with coefficients in R(K) to yield a monic irreducible polynomial with coefficients in K. In an attempt to generalize this concept, Popescu and Zaharescu introduced the notion of lifting with respect to a (K, v)-minimal pair (α, δ) belonging to × . As in the case of usual lifting, a given monic irreducible polynomial Q(y) belonging to R(K(α))[y] gives rise to several monic irreducible polynomials over K which are obtained by lifting with respect to a fixed (K, v)-minimal pair (α, δ). If F, F1 are two such lifted polynomials with coefficients in K having roots θ, θ1, respectively, then it is proved in the present paper that in case (K, v) is a tame field, it is shown that K(θ) and K(θ1) are indeed K-isomorphic.

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 E be a local field, i.e., a field which is complete with respect to a rank one discrete valuation υ (we do not require any finiteness condition on the residue class field of E). Let f(X) be a polynomial in one variable, with coefficients in E. It is well known [4, 6, 9, 11, 13] that the Newton polygon method allows us to gather information about the factorization of f(X). This method consists of attaching to each side S of a Newton polygon of f(X) a factor (not necessarily irreducible) of f(X), the degree of which is the length of the horizontal projection of S.

Nonsingular derivations of modular Lie algebras which have finite multiplicative order play a role in the coclass theory for pro-p groups and Lie algebras. We study the orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of characteristic p > 0. The methods are essentially number-theoretic.

1991 Mathematics subject classification (Amer. Math. Soc): primary 17B50; secondary 12E20.

Let ν be a rank 1 henselian valuation of a field K having unique extension ῡ to an algebraic closure of K. For any subextension L/K of /K, let G (L), Res (L) denote respectively the value group and the residue field of the valuation obtained by restricting ῡ to L. If a∈\K define

We introduce a sequence of polynomials which are extensions of the classic Bernoulli polynomials. This generalization is obtained by using the Bessel functions of the first kind. We use these polynomials to evaluate explicitly a general class of series containing an entire function of exponential type.

Let ν be a valuation of any rank of a field K with value group Gν and f(X)= Xm + alXm−1 + … + am be a polynomial over K. In this paper, it is shown that if (ν(ai)/i)≥(ν(am)/m)>0 for l≤i≤m, and there does not exist any integer r>1 dividing m such that ν(am)/r∈Gν, then f(X) is irreducible over K. It is derived as a special case of a more general result proved here. It generalizes the usual Eisenstein Irreducibility Criterion and an Irreducibility Criterion due to Popescu and Zaharescu for discrete, rank-1 valued fields, (cf. [Journal of Number Theory, 52 (1995), 98–118]).

We give a positive answer to a question of Horst Tietz. A theorem of his that is related to the Mittag-Leffler theorem looks like a duality restult under some locally convex topology on the space of meromorphic functions. Tietz has posed the problem of finding such a topology. It is shown that a topology introduced by Holdguün in 1973 solves the problem. The main tool in the study of this topology is a projective description of it that is derived here. We also argue that Holdgrün's topology is the natural locally convex topology on the space of meromorphic functions.

We study polynomials over an integral domain R which, for infinitely many prime ideals P, induce a permutation of R/P. In many cases, every polynomial with this property must be a composition of Dickson polynomials and of linear polynomials with coefficients in the quotient field of R. In order to find out which of these compositions have the required property we investigate some number theoretic aspects of composition of polynomials. The paper includes a rather elementary proof of ‘Schur's Conjecture’ and contains a quantitative version for polynomials of prime degree.

Kronecker classes of algebraci number fields were introduced by W. Jehne in an attempt to understand the extent to which the structure of an extension K: k of algebraic number fields was influenced by the decomposition of primes of k over K. He found an important link between Kronecker equivalent field extensions and a certain covering property of their Galois groups. This surveys recent contributions of Group Theory to the understanding of Kronecker equivalence of algebraic number fields. In particular some group theoretic conjectures related to the Kronecker class of an extension of bounded degree are explored.

Let K0(x) be a simple transcendental extension of a field K0, υ0 be a valuation of K0 with value group G0 and residue field K0. Suppose is an inclusion of totally ordered abelian groups with [G1: G0] < ∞ such that G is the direct sum of G1 and an infinite cyclic group. It is proved that there exists an (explicitly constructible) valuation υ of K0(x) extending υ0 such that the value group of υ is G and its residue field is k, where k is a given finite extension of k0. This is analogous to a result of Matignon and Ohm [2, Corollary 3.2] for residually non-algebraic prolongations of υ0 to K0(x).

Introduction. Throughout the paper K(x) is a simple transcendental extension of a field K; v is a valuation of K and w is an extension of v to K(x). Also koÍk and GoÍG denote respectively the residue fields and the value groups of the valuations v and w. A well-known theorem conjectured by Nagata asserts that either k is an algebraic extension of feo or k is a simple transcendental extension of a finite extension of ko (cf [4] or [6] or [1, Corollary 2.3]). We prove here an analogous result for the value groups viz. either G/ Go is a torsion group or there exists a subgroup G1 of G containing Go with [G1: Go] > ∞ such that G is the direct sum of G1 and an infinite cyclic group. Incidentally we obtain a description of the valuation w as well as of its residue field in the second case. Thus a characterization of all those extensions w of v to K(x), for which w(K(x)\{0})/Go is not a torsion group, is given. Corresponding to such a valuation w, we define three numbers N, S and T which satisfy the inequality N ≥= ST. This is analogous to the fundamental inequality established by Ohm (cf. [5, 1.2]) for residually transcendental extensions of v to K(x). We also investigate the conditions under which N = ST

The structure of Kronecker class of an extension K: k of algebraic number fields of degree |K: k| ≤ 8 is investigated. For such classes it is shown that the width and socle number are equal and are at most 2, and for those of width 2 the Galois group is given. Further, if |K: k | is 3 or 4, or if 5 ≤ |K: k| ≤ 8 and K: k is Galois, then the groups corresponding to all “second minimal” fields in K are determined.

Let K:= Q(α) be an algebraic number field which is given by specifying the minimal polynomial f(X) for α over Q. We describe a procedure for finding the subfields L of K by constructing pairs (w(X), g(X)) of polynomials over Q such that L= Q(w(α)) and g(X) is the minimal polynomial for w(α). The construction uses local information obtained from the Frobenius-Chebotarev theorem about the Galois group Gal(f), and computations over p-adic extensions of Q.

In this paper we continue our investigations of a construction method for subnear-rings of M(G) proposed by H. Wielandt. For a meromorphic product H, H ⊂ Gk, G finite, we obtain necessary and sufficient conditions for M(G, k, H) to be a near-field.

Let K be an algebraically closed field of characteristic zero, and S a nonempty subset of K such that S Q = Ø and card S < card K, where Q is the field of rational numbers. By Zorn's Lemma, there exist subfields F of K which are maximal with respect to the property of being disjoint from S. This paper examines such subfields and investigates the Galois group Gal K/F along with the lattice of intermediate subfields.