论文标题
关于田地几何理论的注释
A note on geometric theories of fields
论文作者
论文摘要
让$ t $成为一个完整的田野理论，可能具有额外的结构。假设模型理论代数封闭与现场理论代数闭合一致，或者更一般地，模型理论代数封闭具有交换属性。然后，$ t $具有统一的有限性，或者等效地，它消除了量词$ \的存在^\ infty $。因此，从垃圾和Koenigsmann的意义上讲，非常细长的领域与Hrushovski和Pillay的几何场是同一回事。 Modulo一些精美的印刷品，这两个概念也等同于Van Den Dries意义上的代数有限的字段。 从证据中，一个人获得了一条单粒定理用于田地的几何理论：任何无限的可定义集都具有与田地相同的基数。我们研究这是否扩展到可解释的集合。我们表明，正尺寸可解释的集合必须具有与该领域相同的基数，但是零维的可解释集可以具有较小的基数。作为一种应用，我们表明，任何字段的几何理论都有一个无数的模型，只有许多有限的代数扩展。
Let $T$ be a complete theory of fields, possibly with extra structure. Suppose that model-theoretic algebraic closure agrees with field-theoretic algebraic closure, or more generally that model-theoretic algebraic closure has the exchange property. Then $T$ has uniform finiteness, or equivalently, it eliminates the quantifier $\exists^\infty$. It follows that very slim fields in the sense of Junker and Koenigsmann are the same thing as geometric fields in the sense of Hrushovski and Pillay. Modulo some fine print, these two concepts are also equivalent to algebraically bounded fields in the sense of van den Dries. From the proof, one gets a one-cardinal theorem for geometric theories of fields: any infinite definable set has the same cardinality as the field. We investigate whether this extends to interpretable sets. We show that positive dimensional interpretable sets must have the same cardinality as the field, but zero-dimensional interpretable sets can have smaller cardinality. As an application, we show that any geometric theory of fields has an uncountable model with only countably many finite algebraic extensions.
