Algebraic number fields generated by an infinite family of monogenic trinomials

Abstract

For an infinite family of monogenic trinomials Parithmetical invariants of the cubic number field L = Q(θ), generated by a zero θ of P (X), and of its Galois closure N = L( √ d L ) are determined. The conductor f of the cyclic cubic relative extension N/K, where K = Q( √ d L ) denotes the unique quadratic subfield of N , is proved to be of the form 3 e b with e ∈ {1, 2}, which admits statements concerning primitive ambiguous principal ideals, lattice minima, and independent units in L. The number m of non-isomorphic cubic fields L 1 , . . . , Lm sharing a common discriminant d L i = d L with L is determined.