The class number one problem for some totally complex quartic number fields

Abstract

We prove that there are 95 non-isomorphic totally complex quartic fields whose rings of algebraic integers are generated by an algebraic unit and whose class numbers are equal to 1. Moreover, we prove Louboutin's Conjecture according to which a totally complex quartic unit ε u generally generates the unit group of the quartic order Z[ε u ].

The positive discriminant case of Nagell's theorem for certain cubic orders

“…Moreover, in the above listed exceptional cases, u is not a fundamental unit of R. For a quartic analogue of Nagell's theorem the reader is referred to [6]. The main result of the present article (see Theorem 2) answers the above question in the (only remaining) case where f has three distinct real roots i.e.…”

“…Few special-case results of Thomas (from [7]), in which he has determined a complete system of fundamental units of R are then applied to some of our exceptional cases. Our result in the positive discriminant case is: u is a fundamental unit of R except when either max{a, b}, min{a, b} = n 2 , 2n for some integer n 3, or (max{a, b}, min{a, b}) is one of the 7 sporadic pairs: (6,5), (3, −4), (5, −5), (11, −7), (19, −9), (20, −9), (29, −11). Furthermore, u is not a fundamental unit of R in the exceptional cases where (max{a, b}, min{a, b}) is either (6,5) or of the form (n 2 , 2n).…”

“…Our result in the positive discriminant case is: u is a fundamental unit of R except when either max{a, b}, min{a, b} = n 2 , 2n for some integer n 3, or (max{a, b}, min{a, b}) is one of the 7 sporadic pairs: (6,5), (3, −4), (5, −5), (11, −7), (19, −9), (20, −9), (29, −11). Furthermore, u is not a fundamental unit of R in the exceptional cases where (max{a, b}, min{a, b}) is either (6,5) or of the form (n 2 , 2n). For the last six sporadic pairs listed above, our verification (see the remark following Theorem 2) depends on the knowledge of complete sets of fundamental units for the corresponding maximal orders.…”

Determination of the orders generated by a cyclic cubic unit that are Galois invariant

“…It is natural to wonder whether is a fundamental unit of this order. It turns out that the answer is in general yes, and that a fundamental unit of this order can be explicitly given in the rare cases when the answer is no (see [Lou06,Lou08,Lou10,Nag,PL]). Now let be a totally real algebraic cubic unit.…”

On some cubic or quartic algebraic units

“…Here, in Theorem 2, we give a simpler proof of their key result [PL,Theorem 2]. But first, we show in Theorem 1 that their method can also be used to give a simpler proof of [Lou06,Theorem 2], the key result for solving the case of a not totally real cubic algebraic unit.…”