TY - JOUR
AU - Michailov, Ivo
AU - Dimitrov, Ivaylo
AU - Ivanov, Ivan
PY - 2022/03/27
Y2 - 2024/02/27
TI - Noether's Problem for Abelian Extensions of Cyclic p-groups of Nilpotency Class 2
JF - Proceedings of the Bulgarian Academy of Sciences
JA - C. R. Acad. Bulg. Sci.
VL - 75
IS - 3
SE - Mathematics
DO - 10.7546/CRABS.2022.03.01
UR - https://proceedings.bas.bg/index.php/cr/article/view/39
SP - 232-330
AB - <p>Let $$K$$ be a field and $$G$$ be a finite group. Let $$G$$ act on the rational function field $$K(x(g):g\in G)$$ by $$K$$-automorphisms defined by $$g\cdot x(h)=x(gh)$$ for any $$g,h\in G$$. Denote by $$K(G)$$ the fixed field $$K(x(g):g\in G)^G$$. Noether's problem then asks whether $$K(G)$$ is rational over $$K$$. Let $$p$$ be prime and let $$G$$ be a $$p$$-group of exponent $$p^e$$. Assume also that <em>(i)</em> $$K = p>0$$, or <em>(ii)</em> $$K
e p$$ and $$K$$ contains a primitive $$p^e$$-th root of unity. In this paper we prove that $$K(G)$$ is rational over $$K$$ if $$G$$ is any finite $$p$$-group of nilpotency class $$2$$ which is an abelian extension of a cyclic group.</p>
ER -