On nonmonogenic number fields defined by

Abstract

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.

On index divisors and monogenity of certain number fields defined by trinomials $x^6+ax+b$

“…In the remainder of this section, for every prime p, we give sufficient and necessary conditions on a and b so that p is a common index divisor of K. If any one of these conditions holds,then Kis not monogenic. In particular, our proposed results extend those of Jakhar and Kumar given in [25].…”

“…In this paper, for every prime integer p and any number field K defined by an irreducible trinomial F(x) = x 6 + ax + b ∈ Z[x], we characterize when does p is a common index divisor of K. In particular, under any of the mentioned conditions K is not monogenic. In such a way our proposed results extend those of Jakhar and Kumar given in [25].…”

“…In [10], for p = 2, 3, we characterized when p is a common index divisor of any number field generated by a complex root of an irreducible trinomial x 5 + ax 2 + b. In [25], for any number K defined by a trinomial F(x) = x 6 + ax + b ∈ Z[x], Jakhar and Kumar gave some necessary conditions on a and b, which guarantee the non-monogenity of K. Based on the form of the factorization of pZ K , Engstrom calculated ν p (i(K)) for any number field K of degree less than 8 and any prime integer less than 7 [12]. He showed that for sextic number fields, he proved that i(K) = 2 k × 3 l × 5 h for some integers k, l, h such that 0 ≤ k ≤ 8, 0 ≤ l ≤ 3, and h = 0, 1.…”

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For a number field K defined by a trinomailJakhar 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.

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On non-monogenic number fields defined by trinomials of type $x^n +ax^m+b$

“…be the index of the field K. In [9], Davis and Spearman calculated the index of the quartic field defined by x 4 + ax + b. El Fadil gave in [11] necessary and sufficient conditions on a and b so that a rational prime integer p is a common index divisor of number fields defined by x 5 + ax 2 + b. Jakhar and Kumar in [26] gave infinite families of non-monogenic number fields defined by x 6 + ax + b. In [15], Gaál studied the multi-monogenity of sextic number fields defined by trinomials of type x 6 + ax 3 + b.…”