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单词 ExampleOfStronglyMinimal
释义

example of strongly minimal


Let LR be the language of rings.In other words LR has two constant symbols 0,1, one unary symbol -, and two binary function symbols +, satisfying the axioms (identitiesPlanetmathPlanetmathPlanetmath) of a ring. Let T be the LR-theory that includes the field axioms and for each n the formulaMathworldPlanetmathPlanetmath

x0,x1,,xny(¬(1inxi=0)0inxiyi=0)

which expresses that every degree n polynomialMathworldPlanetmathPlanetmath which is non constant has a root. Then any model of T is an algebraically closed field.

One can show that this is a complete theory and has quantifier eliminationMathworldPlanetmath (Tarski).Thus every B-definable subset of any KT is definable by a quantifier free formula in LR(B) with one free variableMathworldPlanetmathPlanetmath y.A quantifier free formula is a Boolean combinationMathworldPlanetmath of atomic formulas.Each of these is of the form inbiyi=0 which defines a finite setMathworldPlanetmath.Thus every definable subset of K is a finite or cofinite set.Thus K and T are strongly minimal

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