Group algebras of finite abelian groups and their applications to combinatorial problems
Group algebras have turned out to be powerful tools for a variety of questions from combinatorics and number theory. For example, in zero-sum problems, the Davenport constant of \(G\), denoted by \(D(G)\), is defined to be the smallest positive integer \(d\) such that every sequence of \(d\) elements in \(G\) contains a nonempty zero-sum subsequence. The problem of finding \(D(G)\) was proposed by H. Davenport in 1966, and he also pointed out that \(D(G)\) is connected to algebraic number theory in the following way. Let \(K\) be an algebraic number field and \(G\) be its class group. Then \(D(G)\) is the maximal number of prime ideals (counting multiplicity) that can occur in the decomposition of an irreducible integer in \(K\). It plays an important role in unique factorization theory in algebraic number theory. The study of \(D(G)\) has attracted a great deal of attention. And the exact value of \(D(G)\) has been determined only for a few classes of groups. Group rings were tried to investigate \(D(G)\). In this paper, the authors considered the following crucial problem which connected with many problems in additive number theory: under what conditions, the sequence \(S=g_1\cdot \ldots \cdot g_l\) over finite abelian group \(G\) satisfies \[ f=(X^{g_1}-a_1)\cdot \ldots \cdot(X^{g_l}-a_l)\neq 0\in R[G] \text{ for all }a_1, \ldots, a_l \in R\setminus \{0\} \] .
scientific article; zbMATH DE number 5566202
scientific article