Trinomial

Rational number

Pure mathematics

Discrete mathematics

Newton polygon

Algebraic number field

Field (mathematics)

Polynomial

Monic polynomial

Mathematics

Mathematical analysis

Combinatorics

Finite field

Journal of Number Theory

Let K=Q(α) be a number field generated by a complex root α of a monic irreducible trinomial F(x)=x6+ax3+b∈Z[x]. There are extensive literature of monogenity of number fields defined by trinomials, Gaál studied the multi-monogenity of sextic number fields defined by x6+ax3+b with a and b two rational integers. Ben Yakkou and El Fadil studied monogenity of some number fields defined by trinomials. In this paper, based on Newton polygon techniques, we deal with the problem of non monogenity of K; when ZK≠Z[η] for every η∈ZK. We give sufficient conditions on a and b for K to be not monogenic. Finally, we illustrate our results by some computational examples.

On non monogenity of certain number fields defined by trinomials x6 + ax3 + b