## Cyclotomy and delta units.(English)Zbl 0789.11060

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.

### MSC:

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

### Citations:

Zbl 0789.11059; Zbl 0652.12004; Zbl 0649.12007
Full Text:

### References:

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