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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
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References:
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