On non-monogenity of certain number fields defined by trinomials $x^4+ax^2+b$

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

“…Hence φ 3 provides two prime ideals of Z K lying above 3 with residue degree 2 each. We conclude that 3Z K = p 11 p 2 12 p 21 p 2 22 p 31 p 2 32 with (3,5), (6, 2)} (mod 9), then N + φ i (F) has a single side of height 1 for every i = 2, 3. Thus there is two prime ideals of Z K lying above 3 with residue degrees 1 and 2 provided by φ 2 and φ 3 respectively.…”

“…In [2], for any quartic number field K defined by a trinomial x 4 + ax + b, Davis and Spearman characterized when p = 2, 3 divides i(K) and showed that i(K) ∈ {1, 2, 3, 6}. In [5], for any quartic number field K defined by a trinomial x 4 + ax 2 + b, El Fadil and Gaál gave necessary and sufficient conditions on a and b, which characterize when a rational prime p divides i(K). In [4], for any rational prime p, El Fadil characterized when p divides the index i(K) of any quintic number field K defined by a trinomial x 5 + ax 2 + b.…”

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

“…For p = 7. Since ∆(F) = 12 12 b 11 − 11 11 a 12 , ν 7 (∆) ≥ 1 if and only if (a, b) ∈ {(0, 0), (1,4), (2,4), (3,4), (4,4), (5,4), (6, 4)} (mod 7). Thanks to the index formula (1.1), 7 can divides the index i(K) only if (a, b) ∈ {(0, 0), (1,4), (2,4), (3,4), (4,4), (5,4), (6,4)} (mod 7).…”

“…(1) For (a, b) ∈ {(1, 4), (2,4), (3,4), (4,4), (5,4), (6,4)} (mod 7), one can easily check that F(x) ≡ φ i1 • φ i2 • φ 2 i3 (mod 7) with deg(φ i1 ) = 7, deg(φ i2 ) = 3, deg(φ i3 ) = 1, and φ i j is irreducible over F 7 for every i = 1, . .…”