## Circular units in a bicyclic field.(English)Zbl 1046.11078

Let $$l$$ be an odd prime, $$K$$ a normal algebraic number field of degree $$l^2$$ over $$\mathbb Q$$ with noncyclic Galois group such that $$l$$ is unramified in $$K$$. Denote the subfields of $$K$$ of degree $$l$$ over $$\mathbb Q$$ with $$K_1, \dots, K_{l+1}$$, the unit group of $$K$$ with $$E$$ and the group of circular units of $$K$$ ($$K_i$$, resp.) with $$C$$ ($$C_i$$, resp.). In Proposition 3.4 the author gives an explicit basis for the subgroup $$B \leq C$$, which is generated by all the $$C_i$$’s. In Section 4 the index $$[C:B]$$ is determined, which depends on the cardinalities of the sets $$P_i$$ ($$1 \leq i \leq l+1$$) of those primes which are ramified in $$K$$, but unramified in $$K_i$$. These ingredients yield a formula for $$[E:C]$$, which is set into relation to the formula of W. Sinnott [Invent. Math. 62, 181–234 (1980; Zbl 0465.12001)] to get an explicit expression for the index $$(R:U)$$ of the “Sinnott module”.

### MSC:

 11R20 Other abelian and metabelian extensions 11R27 Units and factorization

Zbl 0465.12001
Full Text:

### References:

 [1] Dohmae, K., On bases of groups of circular units in some imaginary abelian number fields, J. number theory, 61, 343-364, (1996) · Zbl 0869.11082 [2] Gold, R.; Kim, J., Bases for cyclotomic units, Compos. math., 71, 13-27, (1989) · Zbl 0687.12003 [3] Kučera, R., On bases of the Stickelberger ideal and of the group of circular units of a cyclotomic field, J. number theory, 40, 284-316, (1992) · Zbl 0744.11052 [4] Kučera, R., On the Stickelberger ideal and circular units of a compositum of quadratic fields, J. number theory, 56, 139-166, (1996) · Zbl 0840.11044 [5] Lettl, G., A note on Thaine’s circular units, J. number theory, 35, 224-226, (1990) · Zbl 0705.11064 [6] Sinnott, W., On the Stickelberger ideal and the circular units of an abelian field, Invent. math., 62, 181-234, (1980) · Zbl 0465.12001 [7] Wahington, L., Introduction to cyclotomic fields, graduate texts in mathematics, (1996), Springer New York
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