Class Numbers of Simplest Quartic Fields

Dihedral and cyclic extensions with large class numbers

“…We compute the regulators and prove that they have the largest possible class numbers. Cyclic quartic fields were studied by Lazarus [16], and cyclic sextic fields by Gras [9]. They determined fundamental units.…”

“…It is known that θ 1 , θ 2 and t are independent units where t is the fundamental units of Q( √ t 2 + 16) (See p.10 in [16]). When t is even, we can find t .…”

Elliptic curves associated with simplest quartic fields

“…These fields were studied, among other things, by Gras, who proved that this family is infinite [5]. Later, Lazarus studied their class number [8,9]. More recently, they were studied by Louboutin [10], Kim [7] and Olajos [11].…”

The Brauer–siegel Theorem

“…In particular, if K = K m is the simplest cyclic quartic field associated with the polynomial P m (x) = x 4 − mx 3 − 6x 2 + mx + 1, m 1, m = 3, of discriminant d m = 4∆ 3 m (where the odd part of ∆ m = m 2 + 16 is assumed to be square-free), then we have Reg Km 2,4,5,6,8,9,10,11,15, 24} (see [27], in which a more general problem is in fact solved). Notice that because of much weaker lower bounds for h Km , Lazarus could not solve in [10] and [11] this class number 1 problem for the simplest quartic fields.…”

“…Notice that because of much weaker lower bounds for h Km , Lazarus could not solve in [10] and [11] this class number 1 problem for the simplest quartic fields.…”