\(p\)-ranks of class groups of Witt equivalent number fields
Two algebraic number fields are said to be Witt equivalent if their Witt rings of symmetric bilinear forms are isomorphic (a necessary condition for two fields to be Witt equivalent is that their field degrees over \(\mathbb{Q}\) are equal). Let \(n\) be a prescribed positive integer. In this paper it is proved that, for each prime \(p\) dividing \(n\), every infinite class of Witt equivalent number fields of degree \(n\) contains a field for which the \(p\)-rank of the ideal class group exceeds any given bound. This generalizes the class for \(p=2\) and fields of even degree, for which the result was established previously by the author [Ann. Math. Sil. 12, 53-64 (1998; Zbl 0926.11083)]. Reformulated in terms of Hilbert class fields, the result asserts the existence of a field with infinite Hilbert \(p\)-class field tower in every infinite class of Witt equivalent number fields of degree \(n\). The existence of a field with the properties claimed in the main theorem is established using a theorem of \textit{P. Roquette} and \textit{H. Zassenhaus} [J. Lond. Math. Soc. 44, 31-38 (1969; Zbl 0169.38001)].
infinite Hilbert \(p\)-class field tower
scientific article; zbMATH DE number 1347932