Let q be a prime number and K = Q(θ) be an algebraic number field with θ a root of an irreducible trinomial x 6 +ax+b having integer coefficients. In this paper, we provide some explicit conditions on a, b for which K is not monogenic. As an application, in a special case when a = 0, K is not monogenic if b ≡ 7 mod 8 or b ≡ 8 mod 9. As an example, we also give a non-monogenic class of number fields defined by irreducible sextic trinomials.Throughout the paper, Z K denotes the ring of algebraic integers of an algebraic number field K. For a prime number q and a non-zero m belonging to the ring Z q of q-adic integers, v q (m) will be defined to be the highest power of q dividing m.
Let [Formula: see text] be a prime number and [Formula: see text] be an algebraic number field with [Formula: see text] a root of an irreducible polynomial [Formula: see text] having integer coefficients. In this paper, we provide some explicit conditions involving only [Formula: see text] for which [Formula: see text] is non-monogenic. As an application, in the special case of [Formula: see text] and [Formula: see text], we show that if [Formula: see text] and [Formula: see text] divides [Formula: see text], then [Formula: see text] is not monogenic. We illustrate our results through examples.
Let
$K={\mathbf {Q}}(\theta )$
be an algebraic number field with
$\theta$
a root of an irreducible polynomial
$x^5+ax+b\in {\mathbf {Z}}[x]$
. In this paper, for every rational prime
$p$
, we provide necessary and sufficient conditions on
$a,\,~b$
so that
$p$
is a common index divisor of
$K$
. In particular, we give sufficient conditions on
$a,\,~b$
for which
$K$
is non-monogenic. We illustrate our results through examples.
Let n be a positive integer andn! denote the n-th Taylor polynomial of the exponential function. Let K = Q(θ) be an algebraic number field where θ is a root of f n (x) and Z K denote the ring of algebraic integers of K. In this paper, we prove that for any prime p, p does not divide the index of the subgroup Z[θ] in Z K if and only if p 2 ∤ n!.
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