Annales Mathematiques Africaines


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Cubic Resolvents and Criteria of Monogeneity



François E. TANOE
UFR Mathématiques - Informatique Université Félix Boigny,
22 BP 582 Abidjan 22, Côte d'Ivoire




Mathematics Subject Classification: (MSC 2010) 13A02, 13A15, 13A30, 13B25, 13D40, 13F20, 13H15
Key words : biquadratic fields

Abstract:


A biquadratic fields MATH (in canonical written form), is said to be monogenic i.e. that MATH as a free $\QTR{Bbb}{Z}$-module , of rank $4$, if and only if, the diophantine equation : MATH, , is solvable ( for the known invariants of , $K\ :s=\pm 1$ and the Kronecker MATH where MATH.

In this paper we demonstrate necessary and sufficient conditions among which the following one : $K$ is monogenic of parameters : $u\wedge v=1,$ MATH $(\pm 1,0), $ $(0,\pm 1)$ ) if and only if there exists a primitive element $\alpha $ of $K$ (of Tschirnhauss's type), i.e with irreducible polynomial:

MATH such that its $ps$-cubic resolvent:

MATH admit $3$ integers non square roots, of the type:

MATH, MATH , MATH,

where MATH, and such that:

MATH.

We get similar formulas for the remaining case: MATH or $(0,\ \pm 1)$.

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