Computability and the absolute Galois group of the rational numbers

伽罗瓦绝对群Gal(Q)在数学中是众所周知的。它的元素正是有理数的代数闭包的对称性。然而，在实践中，这一群体特别难以研究。有许多这样的对称性是连续的，其中大多数不能被任何运行任何有限长度程序的计算机（或图灵机）计算出来。然而，数学家经常遇到的对称性本质上总是可计算的——也许是因为它们是这个群体的基础，或者只是因为不可计算的对称性自然更难以研究和处理。这个项目旨在确定可计算对称（作为一个群体）和所有对称的更大群体之间有多大的区别。该研究工作处于逻辑学与数论的交叉领域，有可能引起两方面的兴趣。纽约市立大学研究生中心的研究生将参与本项目。所有实数领域也存在类似的情况：只有可数的实数具有可计算的十进制展开，因此绝大多数实数是不可计算的，然而可计算的实数是日常生活中唯一遇到的实数。在这里，我们知道可计算实数形成了一个与所有实数的全域极其相似的子域，一个具有完全相同一阶性质的初等子域。这笔拨款将资助研究，以试图确定Gal(Q)是否以这种方式类似：可计算对称是否构成完整群的基本子群？（或者，至少，这两者是基本相等的？）如果是这样，那么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体的许多结果，这要容易得多。如果不是，那就表明绝对伽罗瓦群是一个比实数领域更棘手的对象，其不可计算的对称性在某种程度上对其特征至关重要。然而，即便如此，对于相对简单的属性（例如，关于群体的纯粹存在陈述），子群体可能是基本的，在这种情况下，本项目将试图找到子群体停止模仿整个群体的第一层次。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.