On common index divisors and monogenity of certain number fields defined by x⁵ + ax² + b

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

“…(3) (a, b) ∈ {(2, 17), (18, 1)} (mod 32). (4) (a, b) ∈ {(10, 9), (10,5) (14,13), (2,13), (6, 5)} (mod 16) and ν 2 (b + as + s 6 ) = 2ν 2 (a + 6s 5 ) for some integer s for which F(x) is x − s-regular with respect to p = 2.…”

“…For (a, b) ∈ {(14, 13), (6,5), (2,13), (10,5), (10, 9)} (mod 16), we have N + φ 2 (F) has a single side of height 1. By Remark 3, φ 2 provides a unique prime ideal p 21 of Z K lying above 2 with residue degree 2.…”

“…Also in [13], Gaál studied the multi-monogenity of number fields defined by some sextic irreducible trinomials. 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].…”

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.…”

On monogenity of certain number fields defined by $x^{2^r}+ax^m+b$

“…In [7], Davis and Spearman calculated the index of the quartic field defined by x 4 + ax + b. In [17], Gaál studied the monogenity of sextic number fields defined by trinomials of type x 6 + ax 3 + b. El Fadil gave in [10] 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. In [2], Ben Yakkou studied the monogenity of certain number fields defined by x 8 + ax + b.…”