Isomorphism of semisimple group algebras of Abelian groups over a field of second kind (Q2720929)

The paper under review is devoted to the isomorphism problem for group algebras of Abelian groups over a field \(K\) in the semisimple case. Traditionally the investigations are carried out in two different ways depending on the multiplicative properties of \(K\). Let the characteristic of \(K\) be different from the prime \(p\) and let us adjoin a primitive \(p\)-th root of unity for \(p\) odd and a fourth root of unity for \(p=2\). The multiplicative group of the cyclotomic extension obtained has a nontrivial \(p\)-component which is either cyclic (then \(K\) is called a field of the first kind with respect to \(p\)) or isomorphic to \(\mathbb{Z}(p^\infty)\) (then \(K\) is called a field of the second kind with respect to \(p\)). For a fixed field \(K\) of the second kind and an Abelian group \(G\) the author defines a mapping \(\varphi_{KG}\) from the set of all pairs \((F,L)\) of finite dimensional cyclotomic extensions of \(K\), such that \(F\subseteq L\), to the set of cardinals \(\leq|G|\) and proves the following main theorem. If \(KG\) and \(KH\) are semisimple Abelian group algebras over a field \(K\) of the second kind with respect to \(p\), then \(KG\cong KH\) if and only if the functions \(\varphi_{KG}\) and \(\varphi_{KH}\) are equal and \(G\) and \(H\) are isomorphic modulo their torsion components.