On the class number and unit index of simplest quartic fields. (English) Zbl 0719.11073

Lazarus, Andrew J.

Nagoya Math. J. 121, 1-13 (1991).

The simplest quartic fields are defined by adjoining to \({\mathbb{Q}}\) a root of \(X^ 4-tX^ 3-6X^ 2+tX+1\), \(t\in {\mathbb{Z}}^+\), where \(t^ 2+16\) is not divisible by an odd square. This article contains a complete list of simplest quartic fields of even conductor and class number at most two. For even conductor (corresponding to even t), the class number is one if and only if \(t\in \{2,4,6,8,10,24\}\). The class number is two if and only if \(t\in \{12,16,20\}\). The class number of the quadratic subfield is one in all these cases except \(t=16\), where it is two. The author determines the Hasse unit index Q for simplest fields except the case where the fundamental unit of the quadratic subfield has norm -1 and the conductor is an odd composite.

Reviewer: A.J.Lazarus

MSC:

11R29	Class numbers, class groups, discriminants
11R27	Units and factorization
11R16	Cubic and quartic extensions

Keywords:

quartic fields; class number; Hasse unit index

References:

[1]	(1982)
[2]	The Class Number and Cyclotomy of Simplest Quartic Fields (1989)
[3]	pp 285– (1975)
[4]	(1975)
[5]	Publ. math. fasc. 2, Fac. Sci. Besançon (1978)
[6]	DOI: 10.1007/BF01405166 · Zbl 0278.12005
[7]	J. Reine Angew. Math. 284 pp 164– (1976)
[8]	Math. Comp. pp 1137– (1974)
[9]	DOI: 10.1016/0022-314X(85)90055-1 · Zbl 0582.12005
[10]	Math. Comp. 50 pp 543– (1988)
[11]	J. Reine Angew. Math. 275 pp 278– (1975)
[12]	Nagoya Math. J. 112 pp 143– (1988) · Zbl 0629.12004
[13]	Handbook of Mathematical Functions (1964)
[14]	Abh. Deutsche Akad. Wiss 2 pp 1– (1953)
[15]	Number Theory pp 313– (1990)
[16]	(1952)