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


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

Title example of strongly minimal
Canonical name ExampleOfStronglyMinimal
Date of creation 2013-03-22 13:27:46
Last modified on 2013-03-22 13:27:46
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Example
Classification msc 03C45
Classification msc 03C10
Classification msc 03C07
Related topic AlgebraicallyClosed
Defines language of rings