Tame Kummer extensions and Stickelberger conditions (Q789438)

Let \(G\) be a cyclic extension group of order \(m\), \(R\) the ring of integers of an algebraic number field \(K\). A local normal basis theorem of E. Noether implies that the ring of integers of a tame Galois extension \(L\) of \(K\) with Galois group \(G\) represents a class in the class group \(Cl(RG)\) of isomorphism classes of rank one projective \(RG\)-modules. In [Algebraic Number Fields, Proc. Symp. Lond. Math. Soc., Univ. Durham 1975, 561--588 (1977; Zbl 0389.12005)], \textit{L. McCulloh} introduced the problem of characterizing the subgroup \(T(R,G)\) of \(Cl(RG)\) which is generated by rings of integers of tame extensions. He proved that \(T(R,G)=Cl^ 0(RG)^ J\), where \(Cl^ 0(RG)\) is the kernel of the map on class groups induced by the augmentation map from \(RG\) to \(R\), and \(Cl^ 0(RG)^ J\) is the set of elements in \(Cl^ 0(RG)\) which are images under elements of the Stickelberger ideal of \(\mathbb Z\Delta\), \(\Delta =\Aut(G)\) [cf. \textit{S. Lang}, Cyclotomic fields (1978; Zbl 0395.12005), p. 27]. This paper is devoted to the construction of an example \(R\) for which \(T(R,G)\) is not contained in \(Cl^ 0(RG)^ J\) where \(G\) is cyclic of order \(m=\) the square of an odd prime \(p\). The example is the ring of integers of a number field \(K\) containing a primitive \(p^ 2\)-th root of unity, and such that \(Cl(R)\) has a cyclic direct summand of order \(p^ 3\). Such examples exist by a result of \textit{J. Sonn} [Isr. J. Math. 34, 97--105 (1979; Zbl 0433.12005)]. The example shows also that if one multiplies rings of integers of tame extensions of \(K\) with group \(G\) consistent with the Harrison multiplication on the quotient fields, then the map from rings of integers of tame extensions to \(Cl(RG)\) is not a homomorphism.