Algebraic number fields generated by an infinite family of monogenic trinomials

TL;DR

This paper investigates algebraic properties of a specific infinite family of monogenic cubic trinomials, determining invariants of the generated number fields and their Galois closures, including conductors, units, and shared discriminants.

Contribution

It provides explicit descriptions of arithmetical invariants for a family of monogenic cubic fields and their Galois closures, including conductors, units, and classification by discriminant.

Findings

Conductor of the Galois closure is of the form 3^ebb with e in {1,2}

Number of non-isomorphic cubic fields sharing a discriminant is explicitly determined

Properties of primitive ambiguous principal ideals and lattice minima are characterized

Abstract

For an infinite family of monogenic trinomials $P(X) = X^3\pm 3rbX-b$ in $\mathbb{Z}\lbrack X\rbrack$, arithmetical invariants of the cubic number field $L = \mathbb{Q}(\theta)$, generated by a zero $\theta$ of $P(X)$, and of its Galois closure $N = L(\sqrt{d(L)})$ are determined. The conductor $f$ of the cyclic cubic relative extension $N/K$, where $K = \mathbb{Q}(\sqrt{d(L)})$ denotes the unique quadratic subfield of $N$, is proved to be of the form $3^eb$ with $e\in\lbrace 1,2\rbrace$, which admits statements concerning primitive ambiguous principal ideals, lattice minima, and independent units in $L$. The number $m$ of non-isomorphic cubic fields $L_1,\ldots,L_m$ sharing a common discriminant $d(L_i) = d(L)$ with $L$ is determined.