p-进位域上可定义群的一个注记

A Note on Definable Groups in the P-adic Field

利用p-进位域(Qp,+,×,0,1)的模型论方法,研究了域Qp上的可定义群G,当G是局部可交换的,等价于瓖p连通代数群H(Qp),则可定义群G局部同构于H(Qp)且是可交换的,从而得到G是有限可交换的。证明了当G维数为1时,H为代数几何维数为1的连通代数群,p-进位域中一维可定义群是有限可交换的。

Using the model theory method of the p-adic field(Qp,+,×,0,1),the definable group in the field Qp is studied.We prove that when G is locally commutative and is equivalent to the connected algebraic group over H Qp,then G is definably locally isomorphic to H(Qp),and commutative.Thu G is commutative-by-finite.When G is one-dimensional and H is a connected algebraic group of algebraic-geometric dimension 1,one-dimensional groups definable in the p-adics are commutative-by-finite.