# 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 |