Classification and prime decomposition of abelian \(q\)-fields
The author considers abelian \(q\)-fields, \(L\), which are defined to be Galois extensions of the rational number field with Galois group an abelian \(p\)-group. Such a field can be written as the compositum of subfields \(K_1,K_2,\dots, K_n\), where each \(K_i\) is a cyclic \(q\)-field. In the first theorem a formula is given for the absolute discriminant of \(L\). In the second theorem, it is shown that for any prime \(p\), \(L\) can be expressed as a compositum \(K_1K_2\dots K_n\) or \(K_0 K_1 K_2\dots K_n\), where each \(K_i\) is a cyclic \(q\)-field. Moreover, the second expression only occurs when \(q=p=2\). Here \(p\) is totally ramified in \(K_1\) or \(K_0\) and \(K_1\) whenever \(K_0\) is defined. Also, \(p\) is inert in \(K_2\) and splits completely in \(K_3\dots K_n\). On the basis of the latter theorem, the author classifies abelian \(q\)-fields into 4 or 8 types according to the decomposition of a fixed prime \(p\) in \(L\). Only 4 types occur unless \(q=p=2\). No proofs are given here, but the results would appear to follow readily by applying standard Hilbert theory to abelian extensions.
scientific article; zbMATH DE number 1053789