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# 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

## Mathematics Subject Classification

03C45*no label found*03C10

*no label found*03C07

*no label found*

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