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In this article, we establish a connection between key polynomials over a residually transcendental prolongation of a henselian valuation on a field K and distinguished pairs. We also derive a new ...

Let $K=\mathbb{Q}(\sqrt[n]{a})$ be an extension of degree $n$ of the field $\Q$ of rational numbers, where the integer $a$ is such that for each prime $p$ dividing $n$ either $p\nmid a$ or the highest power of $p$ dividing $a$ is coprime to $p$; this condition is clearly satisfied when $a, n$ are coprime or $a$ is squarefree. The present paper gives explicit construction of an integral basis of $K$ along with applications. This construction of an integral basis of $K$ extends a result proved in [J. Number Theory, {173} (2017), 129-146] regarding periodicity of integral bases of $\mathbb{Q}(\sqrt[n]{a})$ when $a$ is squarefree.

Let v be a Krull valuation of a field K with valuation ring R v and K 1 , K 2 be finite separable extensions of K which are linearly disjoint over K. Assume that the integral closure of R v in the composite field K 1 K 2 is a free R v -module. For a given pair of prolongations v 1 , v 2 of v to K 1 , K 2 respectively, it is shown that there exists a unique prolongation w of v to K 1 K 2 which extends both v 1 , v 2 . Moreover with S i as the integral closure of R v in K i , if the ring S 1 S 2 is integrally closed and the residue field of v is perfect, then f ( w / v ) = f ( v 1 / v ) f ( v 2 / v ) , where f ( v ′ / v ) stands for the degree of the residue field of a prolongation v ′ of v over the residue field of v. As an application, it is deduced that if K 1 , K 2 are algebraic number fields which are linearly disjoint over K = K 1 ∩ K 2 , then the number of prime ideals of the ring A K 1 K 2 of algebraic integers of K 1 K 2 lying over a given prime ideal ℘ of A K equals the product of the numbers of prime ideals of A K i lying over ℘ for i = 1 , 2 .

Let K = ℚ(θ) be an extension of the field ℚ of rational numbers where θ satisfies an irreducible polynomial xp − a of prime degree belonging to ℤ[x]. In this paper, we give explicilty an integral basis for K using only elementary algebraic number theory. Though an integral basis for such fields is already known (see [Trans. Amer. Math. Soc., 11 (1910), 388–392)], our description of integral basis is different and slightly simpler. We also give a short proof of the formula for discriminant of such fields.

Let $K=\mathbb{Q}(\sqrt[n]{a})$ be an extension of degree $n$ of the field $\Q$ of rational numbers, where the integer $a$ is such that for each prime $p$ dividing $n$ either $p\nmid a$ or the highest power of $p$ dividing $a$ is coprime to $p$; this condition is clearly satisfied when $a, n$ are coprime or $a$ is squarefree. The paper contains an explicit formula for the discriminant of $K$ involving only the prime powers dividing $a,n$.

It is well known that if K 1 , K 2 are algebraic number fields with coprime discriminants, then the composite ring A K 1 A K 2 is integrally closed and K 1 , K 2 are linearly disjoint over the field of rationals, A K i being the ring of algebraic integers of K i . In an attempt to prove the converse of the above result, in this paper we prove that if K 1 , K 2 are finite separable extensions of a valued field ( K , v ) of arbitrary rank which are linearly disjoint over K = K 1 ∩ K 2 and if the integral closure S i of the valuation ring R v of v in K i is a free R v -module for i = 1 , 2 with S 1 S 2 integrally closed, then the discriminant of either S 1 / R v or of S 2 / R v is the unit ideal. We quickly deduce from this result that for algebraic number fields K 1 , K 2 linearly disjoint over K = K 1 ∩ K 2 for which A K 1 A K 2 is integrally closed, the relative discriminants of K 1 / K and K 2 / K must be coprime.

For positive integers k and n with k ⩽ n − 1 , let P n , k ( x ) denote the polynomial ∑ j = 0 k ( n j ) x j , where ( n j ) = n ! j ! ( n − j ) ! . In 2011, Khanduja, Khassa and Laishram proved the irreducibility of P n , k ( x ) over the field Q of rational numbers for those n , k for which 2 ≤ 2 k ≤ n ( k + 1 ) 3 . In this paper, we extend the above result and prove that if 2 ≤ 2 k ≤ n ( k + 1 ) e + 1 for some positive integer e and the smallest prime factor of k is greater than e, then there exists an explicitly constructible constant C e depending only on e such that the polynomial P n , k ( x ) is irreducible over Q for k ≥ C e .

Let v be a Krull valuation of a field with valuation ring R v . Let θ be a root of an irreducible trinomial F ( x ) = x n + a x m + b belonging to R v [ x ] . In this paper, we give necessary and sufficient conditions involving only a , b , m , n for R v [ θ ] to be integrally closed. In the particular case when v is the p -adic valuation of the field Q of rational numbers, F ( x ) ∈ Z [ x ] and K = Q ( θ ) , then it is shown that these conditions lead to the characterization of primes which divide the index of the subgroup Z [ θ ] in A K , where A K is the ring of algebraic integers of K . As an application, it is deduced that for any algebraic number field K and any quadratic field L not contained in K , we have A K L = A K A L if and only if the discriminants of K and L are coprime.

Guardia, Montes and Nart generalized the well-known method of Ore to find complete factorization of polynomials with coefficients in finite extensions of p-adic numbers using Newton polygons of higher order (cf. [Trans. Amer. Math. Soc. 364 (2012), 361–416]). In this paper, we develop the theory of higher order Newton polygons for polynomials with coefficients in henselian valued fields of arbitrary rank and use it to obtain factorization of such polynomials. Our approach is different from the one followed by Guardia et al. Some preliminary results needed for proving the main results are also obtained which are of independent interest.

Let AK denote the ring of algebraic integers of an algebraic number field K=Q(θ) where the algebraic integer θ has minimal polynomial F(x)=xn+axm+b over the field Q of rational numbers with n=mt+u, t∈N, 0≤u≤m−1. In this paper, we characterize those primes which divide the discriminant of F(x) but do not divide [AK:Z[θ]] when u=0 or u divides m; such primes p are important for explicitly determining the decomposition of pAK into a product of prime ideals of AK in view of the well known Dedekind theorem. As a consequence, we obtain some necessary and sufficient conditions involving only a, b, m, n for AK to be equal to Z[θ].

We state a mild generalization of the classical Schonemann irreducibility criterion in ℤ[x] and provide an elementary proof.

Let [Formula: see text] be a discrete valuation ring with maximal ideal [Formula: see text] and [Formula: see text] be the integral closure of [Formula: see text] in a finite separable extension [Formula: see text] of [Formula: see text]. For a maximal ideal [Formula: see text] of [Formula: see text], let [Formula: see text] denote respectively the valuation rings of the completions of [Formula: see text] with respect to [Formula: see text]. The discriminant satisfies a basic equality which says that [Formula: see text]. In this paper, we extend the above equality on replacing [Formula: see text] by the valuation ring of a Krull valuation of arbitrary rank and completion by henselization. In the course of proof, we prove a generalization of the well-known weak Approximation Theorem which is of independent interest as well.