Cyclotomy and delta units. (English) Zbl 0789.11060

Lazarus, Andrew J.

Math. Comput. 61, No. 203, 295-305 (1993).

Let \(p\) be an odd prime and let \(\zeta\) be a primitive \(p\)th root of unity. Let \(K \subseteq \mathbb{Q} (\zeta)\) be a field with \([K:\mathbb{Q} (\zeta)]=e\). Let \(\eta=\text{Tr} (\zeta)\) be the trace of \(\zeta\) from \(\mathbb{Q} (\zeta)\) to \(K\). Then \(\eta\) and its Galois conjugates are Gaussian periods. E. Lehmer [Math. Comput. 50, 535-541 (1988; Zbl 0652.12004)] discovered that, in the cases \(e=2,3,4\), and 5, certain families of fields contain units that are of the form \(\eta+c\), with \(c \in \mathbb{Z}\). Such units are called translation units. In the cases \(e=2,3,4\), R. Schoof and the reviewer [Math. Comput. 50, 543-556 (1988; Zbl 0649.12007)] showed that if \([K:\mathbb{Q}]=e\) and \(K\) contains a unit of the form \(\eta+c\) (which must be assumed to be of positive norm if \(e=4)\), then \(K\) belongs to one of these families. More recently, D. H. Lehmer and E. Lehmer [Math. Comput. 61, 313-317 (1993), see the review above] studied differences \(\delta=\text{Tr} (\zeta)- \text{Tr} (\zeta^ g)\), where \(g\) is a primitive root mod \(p\), and showed that integer translates of \(\delta\) may sometimes be used to produce units.
Call a unit of the form \(\delta \pm 1\) a delta unit, and a unit of the form \(\delta+c\), with \(c \in \mathbb{Z}\), a generalized delta unit. The author proves
Theorem 1: If \(K\) is a cyclic cubic field of prime conductor \(p\), the following are equivalent: (i) \(K\) is a “simplest cubic field,” defined by a polynomial \(X^ 3-aX^ 2-(a+3) X-1\) with \(p=a^ 2+3a+9\), (ii) \(K\) contains a translation unit, (iii) \(K\) contains a delta unit, (iv) \(K\) contains a generalized delta unit.
Theorem 2: If \(K\) is a real cyclic quartic field of prime conductor \(p\), the following are equivalent: (i) \(K\) is a “simplest quartic field,” defined by a polynomial \(X^ 4-aX^ 3-6X^ 2+ aX+1\) with \(p=a^ 2+16\), (ii) \(K\) contains a translation unit of positive norm, (iii) \(K\) contains a delta unit, (iv) \(K\) contains a generalized delta unit of positive norm.
In the case \(e=5\), the existence of generalized delta units can depend on the choice of the primitive root \(g\). The author investigates this phenomenon.

Reviewer: L.Washington (College Park)

MSC:

11R18	Cyclotomic extensions
11R16	Cubic and quartic extensions
11R27	Units and factorization

Keywords:

cyclotomy; Gaussian periods; translation units; delta unit; generalized delta unit; cyclic cubic field; real cyclic quartic field

Citations:

Zbl 0789.11059; Zbl 0652.12004; Zbl 0649.12007

References:

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