# example of strongly minimal

Let $L_{R}$ be the language of rings. In other words $L_{R}$ has two constant symbols $0,1$, one unary symbol $-$, and two binary function symbols $+,\cdot$ satisfying the axioms (identities) of a ring. Let $T$ be the $L_{R}$-theory that includes the field axioms and for each $n$ the formula

 $\forall x_{0},x_{1},\ldots,x_{n}\exists y(\lnot(\bigwedge_{1\leq i\leq n}x_{i}% =0)\rightarrow\sum_{0\leq i\leq n}x_{i}y^{i}=0)$

which expresses that every degree $n$ polynomial 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 elimination (Tarski). Thus every $B$-definable subset of any $K\models T$ is definable by a quantifier free formula in $L_{R}(B)$ with one free variable $y$. A quantifier free formula is a Boolean combination of atomic formulas. Each of these is of the form $\sum_{i\leq n}b_{i}y^{i}=0$ which defines a finite set. Thus every definable subset of $K$ is a finite or cofinite set. Thus $K$ and $T$ are strongly minimal

Title example of strongly minimal ExampleOfStronglyMinimal 2013-03-22 13:27:46 2013-03-22 13:27:46 CWoo (3771) CWoo (3771) 7 CWoo (3771) Example msc 03C45 msc 03C10 msc 03C07 AlgebraicallyClosed language of rings