On integral representations over cyclotomic fields (Q678395)

The following generalization of the second author's thesis [\textit{O. Neiße}, Augsburger Mathematisch-Naturwissenschaftliche Schriften. 3 (Wißner Verlag, 1995; Zbl 0826.20004)] is proved: Let \(G\) be a finite solvable group and \(\chi\) an irreducible complex character of \(G\) with odd degree. Then there exists a representation of \(G\) by matrices over \(\mathbb{Z}[\zeta_f]\) belonging to \(\chi\). Here \(f\) denotes the conductor of the number field \(\mathbb{Q}(\chi)=\mathbb{Q}(\{\chi(g)\mid g\in G\})\) and \(\zeta_f\) a primitive \(f\)-th root of unity. It is not known whether this result generalizes to nonsolvable groups -- at least no counterexample is known so far.