AFRICA MATHEMATICS ANNALS
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Contents > Past issues > Volume 5
On 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
(in canonical written form), is said to be monogenic i.e. that
as a free
-module
, of rank
,
if and only if, the diophantine equation :
,
, is solvable ( for the known invariants of ,
and the Kronecker
where
.
In this paper we demonstrate necessary and sufficient conditions among which
the following one :
is monogenic of parameters :
) if and only if there exists a primitive element
of
(of Tschirnhauss's type), i.e with irreducible polynomial:
such that its
-cubic
resolvent:
admit
integers non square roots, of the type:
,
,
,
where
,
and such that:
.
We get similar formulas for the remaining case:
or
.