Units in totally complex \(S_3\) fields
Let \(B\) be a totally complex number field, Galois over the rational field \(\mathbb Q\), with Galois group isomorphic to the symmetric group \(S_3\). Denote by \(F_1,F_2,F_3\) the cubic subfields and by \(k\) the complex quadratic subfield of \(B\). The author determines the inequivalent representations of \(S_3\) acting on \(U\), the group of units of \(B\), and gives arithmetic criteria for deciding which representation occurs for a given field. These criteria are stated by the use of the concept of quasi-principal factorization (QPF): in a cubic extension \(L/K\), there is a QPF if there exists a principal ideal \((A)\) of \(K\) such that \((A)\) is the cube of a principal ideal of \(L\) but no generator of \((A)\) in \(K\) is the cube of an element of \(L\). An example of the results: Let \(U_0\) be the subgroup of \(U\) generated by the roots of unity in \(B\) and the fundamental units \(\eta_i\) of \(F_i\) \((i = 1,2,3)\), normalized in a natural way. Then \(U=U_0\) iff there is a QPF in \(B/k\) but not in \(F_1/\mathbb Q\). As a result of the theory, a computational procedure is given for determining a pair of fundamental units of \(B\), given a fundamental unit of \(F_1\).
scientific article; zbMATH DE number 3595281