Properties of forking in ω-free pseudo-algebraically closed fields
Properties of forking in ω-free pseudo-algebraically closed fields
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DOI：
10.2178/jsl/1190150143
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发表时间：
2002-09
影响因子：
0.6
通讯作者：
Z. Chatzidakis
中科院分区：
文献类型：
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作者：
Z. Chatzidakis
伪代数闭域（以下称为PAC）的研究始于J. Ax对有限域和伪有限域的研究。他证明了有限域理论的无限模型正是与绝对伽罗瓦群同构的完美PAC域，并给出了它们一阶理论的初等不变量，从而证明了有限域理论的可决性。Ax的结果随后被M. Jarden和U. Kiehne[21]和Jarden[21]扩展到更大的PAC领域。关于PAC场理论的最后定论是由G. Cherlin， L. van den Dries和A. Macintyre提出的，参见Ju的结果。Ershov[13][14]。设K为PAC字段。那么K的初等理论完全由以下数据决定：•K的绝数域的同构类型（素域上元素代数的K的子域）。•用一种合适的ω排序语言，给出了伽罗瓦群逆系统（L/K）的一阶理论，其中L遍历K的所有有限伽罗瓦扩展。他们还证明了PAC域的理论是不可判定的，通过证明任何图都可以编码在某个PAC域的绝对伽罗瓦群中。事实证明，绝对的伽罗瓦组控制着PAC字段的大部分行为。我将在下面给出一些例子来说明这种现象。
The study of pseudo-algebraically closed fields (henceforth called PAC) started with the work of J. Ax on finite and pseudo-finite fields [1]. He showed that the infinite models of the theory of finite fields are exactly the perfect PAC fields with absolute Galois group isomorphic to , and gave elementary invariants for their first order theory, thereby proving the decidability of the theory of finite fields. Ax's results were then extended to a larger class of PAC fields by M. Jarden and U. Kiehne [21], and Jarden [19]. The final word on theories of PAC fields was given by G. Cherlin, L. van den Dries and A. Macintyre [10], see also results by Ju. Ershov [13], [14]. Let K be a PAC field. Then the elementary theory of K is entirely determined by the following data: • The isomorphism type of the field of absolute numbers of K (the subfield of K of elements algebraic over the prime field). • The degree of imperfection of K. • The first-order theory, in a suitable ω-sorted language, of the inverse system of Galois groups al(L/K) where L runs over all finite Galois extensions of K. They also showed that the theory of PAC fields is undecidable, by showing that any graph can be encoded in the absolute Galois group of some PAC field. It turns out that the absolute Galois group controls much of the behaviour of the PAC fields. I will give below some examples illustrating this phenomenon.