Computability and the absolute Galois group of the rational numbers

绝对伽罗华群Gal(Q)在整个数学中是众所周知的。它的元素正好是有理数的代数闭包的对称性。然而，在实践中，这一群体尤其难以研究。有连续体-许多这样的对称性，其中大多数不能由任何运行任何有限长度程序的计算机(或图灵机)计算。然而，数学家经常遇到的对称性基本上总是可计算的--也许是因为这些是群的基础，或者仅仅是因为不可计算的对称性自然更难检查和处理。这个项目的目的是确定可计算的对称(作为一个群)和所有对称中更大的群之间到底有多大的差异。这项研究工作位于逻辑和数论的交界处，可能会引起两个社区的兴趣。纽约州立大学研究生中心的研究生将参与这个项目。所有实数的领域也存在类似的情况：只有可计数的许多实数具有可计算的小数展开，因此绝大多数实数是不可计算的，但可计算的实数是日常生活中唯一遇到的。这里，我们知道，可计算的实数形成一个与所有实数的全域极其相似的子域，一个具有完全相同的一阶性质的初等子域。这笔拨款将用于研究，试图确定Gal(Q)是否以这种方式相似：可计算的对称是否形成整个群的初等子群？(或者，至少，这两者在本质上是等价的？)如果是这样的话，数学家应该能够通过检查可计算的对称性来确定关于整个群的许多结果，这些对称性要容易得多。如果不是，这将表明绝对伽罗瓦群是一个比实数领域更棘手的对象，因为它的不可计算对称性在某种程度上对它的性质是必不可少的。然而，即使在那时，子组可能是相对简单的属性的基础(例如，关于组的纯粹存在主义的陈述)，在这种情况下，本项目将试图找到子组停止模仿整个组的第一个级别。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The absolute Galois group Gal(Q) is well-known throughout mathematics. Its elements are precisely the symmetries of the algebraic closure of the rational numbers. In practice, though, this group is particularly difficult to study. There are continuum-many of these symmetries, most of which cannot be computed by any computer (or Turing machine) running any finite-length program whatsoever. However, the symmetries that mathematicians encounter on a regular basis are essentially always computable -- perhaps because these are fundamental to the group, or perhaps just because noncomputable symmetries are naturally more difficult to examine and work with. This project aims to determine just how much difference there is between the computable symmetries (as a group) and the larger group of all symmetries. The research work lies at the interface of logic and number theory and is likely to attract the interest of both communities. Graduate students from CUNY Graduate Center will participate in this project. An analogous situation exists with the field of all real numbers: only countably many real numbers have computable decimal expansions, so the vast majority of real numbers are noncomputable, yet the computable ones are the only ones ever encountered in daily life. Here, it is known that the computable real numbers form a subfield extremely similar to the full field of all real numbers, an elementary subfield with exactly the same first-order properties. This grant will fund research to attempt to determine whether Gal(Q) is analogous in this way: do the computable symmetries form an elementary subgroup of the full group? (Or, at a minimum, are the two elementarily equivalent?) If so, then mathematicians should be able to determine many results about the full group just by examining the computable symmetries, which are far more accessible. If not, that would suggest that the absolute Galois group is a thornier object than the field of real numbers, with its noncomputable symmetries somehow essential to its character. However, even then, it is possible that the subgroup might be elementary for relatively simple properties (e.g., purely existential statements about the group), in which case this project will attempt to find the first level at which the subgroup stops imitating the full group.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.