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[parent] example of strongly minimal (Example)

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

$\displaystyle \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



"example of strongly minimal" is owned by CWoo. [ full author list (2) | owner history (1) ]
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See Also: algebraically closed

Also defines:  language of rings

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Cross-references: strongly minimal, cofinite, finite, finite set, atomic formulas, combination, Boolean, free variable, quantifier free formula, definable, subset, quantifier elimination, complete theory, algebraically closed, root, polynomial, degree, formula, field, ring, identities, axioms, function symbols, binary, unary, constant symbols

This is version 4 of example of strongly minimal, born on 2003-02-12, modified 2007-08-10.
Object id is 4030, canonical name is ExampleOfStronglyMinimal.
Accessed 2294 times total.

Classification:
AMS MSC: 03C07 (Mathematical logic and foundations :: Model theory :: Basic properties of first-order languages and structures)
 03C10 (Mathematical logic and foundations :: Model theory :: Quantifier elimination, model completeness and related topics)
 03C45 (Mathematical logic and foundations :: Model theory :: Classification theory, stability and related concepts)
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