On index divisors and monogenity of certain septic number fields defined by x⁷ + ax³ + b

On index divisors and monogenity of certain number fields defined by x12+axm+b

“…), (0, 7), (3, 1), (3,4), (3,7), (6, 1), (6, 4), (6, 7)} (mod 9). 12 , thanks to the index formula (1.1), p is the unique rational prime candidate to divide (Z K : Z[α]).…”

“…In [6], for any sextic number field defined by a trinomial x 6 + ax 5 + b, we characterized ν p (i(K)) for p = 2, 3, 5 and showed that i(K) ∈ {1, 2, 3, 4, 6, 12}. Also in [7], for every rational prime p, we characterized ν p (i(K)) for any septic number field defined by a trinomial x 7 + ax 3 + b and proved that i(K) ∈ {1, 2, 3, 4, 6, 12}. In this paper, for any number field K defined by a monic irreducible trinomial F(x) = x 12 + ax m + b ∈ Z[x], we give sufficient conditions on a, b, and m so that the index i(K) is not trivial.…”

“…Hence ν 3 (i(K)) = 0. (iv) If d = 4, then N φ (F) = S 1 has a single side joining (0, 4) and (12,0) with 7 ) and a ≡ 0 (mod 3 6 ), then N φ (F) = S 1 has a single side joining (0, 6) and (12, 0) with…”

“…(a) If a ≡ ±3 6 (mod 3 7 ), then N + 2,i (F) = T i has a single side joining (0, 13) and (3,12) for every i = 1, 2. Thus 3Z K = p 6 1 p 6 2 with residue degree 1 each.…”

On index divisors and non-monogenity of certain quintic number fields defined by x5 + axm + bx + c

“…In [4], for any rational prime p, El Fadil characterized when p divides the index i(K) for any quintic number field K defined by a trinomial x 5 + ax 2 + b. In [7], for every rational prime p, we characterized ν p (i(K)) for any septic number field defined by a trinomial x 7 + ax 3 + b.…”