On integral bases and monogeneity of pure sextic number fields with non-squarefree coefficients

On common index divisors and monogenity of certain number fields defined by $x^5+ax+b$

“…(1) For F(x) = x 5 + 3x 2 + 48u with u an odd integer and 3 does not divide u, we have (5) For a ≡ ±4 (mod 9), 3 is not a common index of K for every value of b. (6) Assume that a ≡ 7 (mod 3 4 ) and b ≡ a 5 − a 3 ≡ 21 (mod 3 4 ).…”

“…Thus 3 is a common index divisor of K. If a ≡ 7 (mod 3 5 ) and b ≡ 183 (mod 3 5 ) , then N + φ 2 (F) = S 21 + S 22 has two sides joining (0, 4), (2, 1), and (3, 0), with R λ 21 (F)(y) = −(y 2 + 1), which is irreducible over F 3 . Thus 3 is not a common index divisor of K. For a ≡ 88+3 6 A and b ≡ a 5 −a 3 (mod 3 6 ), we have F(x) ≡ φ 6 2 +(729A+1747)φ 5 2 + (1458A + 895)φ 4 2 + (729A + 60)φ b ≡ 5934 (mod 3 8 ), then N + φ 2 (F) has three sides joining (0, v), (1,4), (2, 1), and (3, 0). Thus φ 2 provides three prime ideals of Z K lying above 3 with residue degree 1 each, and so 3 is a common index divisor of K. (7) If a ≡ 11 (mod 3 5 ) and b ≡ 69 (mod 3 6 ) , then N + φ 2 (F) = S 21 + S 22 has two sides joining (0, v), (1,2), and (3, 0), with v ≥ 4 and R λ 22 (F)(y) = y 2 + y + 1 = (y − 1) 2 .…”

“…Thus 3 is not a common index divisor of K. For a ≡ 88+3 6 A and b ≡ a 5 −a 3 (mod 3 6 ), we have F(x) ≡ φ 6 2 +(729A+1747)φ 5 2 + (1458A + 895)φ 4 2 + (729A + 60)φ b ≡ 5934 (mod 3 8 ), then N + φ 2 (F) has three sides joining (0, v), (1,4), (2, 1), and (3, 0). Thus φ 2 provides three prime ideals of Z K lying above 3 with residue degree 1 each, and so 3 is a common index divisor of K. (7) If a ≡ 11 (mod 3 5 ) and b ≡ 69 (mod 3 6 ) , then N + φ 2 (F) = S 21 + S 22 has two sides joining (0, v), (1,2), and (3, 0), with v ≥ 4 and R λ 22 (F)(y) = y 2 + y + 1 = (y − 1) 2 . For s = −a + 3 and φ 2 = x − s, we have N + φ 2 (F) = S 21 + S 22 + S 23 has three sides joining (0, v), (1,3), (2,1), and (3, 0) with v ≥ 6.…”

“…Based on prime ideal factorization, we showed in [7] that for any square free rational integer m ±1, if m ≡ 1 (mod 4) or m ≡ 1 (mod 9), then the pure sextic field Q( 6 √ m) is not monogenic. We also studied in [6] the monogenity of Q( 6√ m) with m ±1 an integer, which is not necessarily square free. In [21,22], Kunduja et al studied the integral closedness of some number fields defined by trinomials.…”

On power integral bases of certain pure number fields defined by $X^{60}-m$

“…While Gaál's and Remete's techniques are based on the index calculation, Nakahara's methods are based on the existence of power relative integral bases of some special sub-fields. In [7,8,10,12,11,9], El Fadil et al used Newton polygon techniques to study the monogeneity of the pure number fields of degrees 6, 12, 18, 24, and 36. In this paper, Our purpose is for a square free integer m ±1 and F(x) = x 60 − m is an irreducible polynomail over Q, to study the monogeneity of the number field K = Q(α) generated by a complex root α of F(x).…”