Application Of Model Theory In Algebra

Model theory is widely used in many fields of mathematics,especially in the field of algebra.This article first introduces the main results of the application of model theory in mathematics-related fields,including the application in algebra,Secondly,the related properties of model theory in the algebraically closed fields and p-adic field are discussed.The main contents are as follows:Explore the model theory on the algebraically closed fields,firstly prove the completeness,decidability,and quantifier elimination of the algebraically closed fields.Secondly,by introducing the Zariski closed set,constitutable set and Morley rank related concept theory.It is proved that if it is a non-empty constructable set,then its Morley rank is equal to its dimension.Finally,the relationship between the algebraically closed fields eliminate imaginaries and the equivalence relation is proved,and the related theory of the algebraically closed fields eliminate imaginaries is proved.Using the model theory method of the field(Qp,+,x,0,1),the definable group in the p-adic field is studied.We prove here that when G is locally commutative,equivalently the(connected)algebraic group H over Qp such that G is definably locally isomorphic to H(Qp),is commutative,the G is commutative-by-finite.When G is one-dimensional(in the sense of p-adic dimension)then H will be a connected algebraic group of algebraic-geometric dimension 1,so commutative.Thus we deduce from our results that one-dimensional groups definable in the p-adics are commutative-by-finite.