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

$$\forall x_0,x_1,\mathrm{\dots },x_n\exists y(\mathrm{\neg }(\underset{1\le i\le n}{\bigwedge }{x}_i=0)\to \sum _{0\le i\le n}{x}_i{y}^i=0)$$

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 $K\vDash T$ 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 ${\sum }_{i\le n}{b}_i{y}^i=0$ 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
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