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AbstractIn this paper, we calculate the index of any septic number field K generated by a root α of a monic irreducible trinomial F(x) = x7 + ax4 + b ∈ ℤ[x]. Our approach is based on Engstrom’s results and the factorization of any rational prime in K. In such a way we give a complete answer of Problem 22 of Narkiewicz ([28]) for this family of number fields. As an application of our results, if i(K) ≠ 1, then K is not monogenic. Also, we give generators of power integral bases in some cases where i(K) = 1. Our results are illustrated by some computational examples. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic 0, and let f (x) = x 8 + a x 6 + b x 4 + a x 2 + 1 ∈ F [ x ] be irreducible with Galois group Gal (f) . We extend results in the literature to give a complete and elementary classification of Gal (f) , identified as a transitive subgroup of S8 up to conjugacy. The approach involves testing whether a small number of elements in F are perfect squares. As an application, we give one-parameter families of polynomials for each of the six possible Galois groups. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to calculate the index of any number field K generated by a root of an irreducible trinomial x 4 + a x + b ∈ Z [ x ] . Our approach is based on Engstrom's results and the factorization of 2 Z K and 3 Z K . In particular, we reformulate the Davis and Spearman's results. Namely, for every prime integer p, we evaluate ν p (i (K)) . The existence of a common index divisor of K guaranties the non monogenity of K. [ABSTRACT FROM AUTHOR]

For a number field K defined by a trinomial F (x) = x6 + ax + b ∈ ℤ[x], Jakhar and Kumar gave some necessary conditions on a and b, which guarantee the non-monogenity of K [25]. In this paper, for every prime integer p, we characterize when p is a common index divisor of K. In particular, if any one of these conditions holds, then K is not monogenic. In such a way our proposed results extend those of Jakhar and Kumar. [ABSTRACT FROM AUTHOR]

In this article, we deal with the problem of monogenity of quintic number fields defined by monic irreducible quadrinomials F (x) = x 5 + a x m + b x + c ∈ Z [ x ] with m = 2, 3, 4. We give sufficient conditions on a, b, c, and m so that the index of the field K is nontrivial and we evaluate ν p (i (K)) in each case. In particular, if i (K) ≠ 1 , then K is not monogenic. Our results are illustrated by computational examples. [ABSTRACT FROM AUTHOR]

In this paper, for any septic number field K generated by a root α of a monic irreducible trinomial F (x) = x 7 + a x 3 + b ∈ Z [ x ] , we describe all prime power divisors of the index of K answering Problem 22 of Narkiewicz [ 26]. In particular, if i (K) ≠ 1 , then K is not mongenic. We illustrate our results by some computational examples. [ABSTRACT FROM AUTHOR]

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 invariant for an element θ belonging to an algebraic closure of K, which is of independent interest as well. [ABSTRACT FROM AUTHOR]

In this paper, we are going to exploit the irreducibility of the composite polynomial defined on the same field to prove a New and useful result linking the index-ideal in Tower extensions allowing to study the relative monogeneity of extensions of a numbers fields. Being done, we will illustrate this result for quartic and in sextic extensions. [ABSTRACT FROM AUTHOR]

We give an infinite family of intersective polynomials with Galois groupA4, the alternating group on four letters. [ABSTRACT FROM AUTHOR]