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Contents > Past issues > Volume 5

On Cubic Resolvents and Criteria of Monogeneity

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

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 .