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

Kraemer, Pavel

J. Number Theory 105, No. 2, 302-321 (2004).

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”.

Reviewer: Günter Lettl (Graz)

MSC:

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

Keywords:

cyclotomic units; class number formula; Sinnott module; Sinnott index formula

Citations:

Zbl 0465.12001

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