论文标题
从自然数到主要领域和有限领域
From natural numbers to prime fields and finite fields
论文作者
论文摘要
这本书的目的是通过从自然数字到主要领域和有限领域的旅程向读者介绍代数的美丽，并带有一些弯路。 许多书籍专门用于从自然数中构建这些领域。也许第一个是[Landau 1930]，另请参见[Ebbinghaus et.al。 [1991]，[LAY 2005]，其中的参考文献。在此过程中，重要的一步是理性数字领域的构建。事实证明，该领域是唯一的（从适当的意义上）无限的素数。在[Loonstra 1972]的阅读中，我决定写一本基本独立的书，不仅专门针对理性数字的构建，而且是所有主要领域的构建，更有趣的是，更有趣的是，所有有限领域的存在和“独特”。假定只有对基本集合理论的知识和对数学推理的一些熟悉程度（例如，请参见[11，第1和第2章]）。
The aim of this book is to introduce the reader to the beauty of Algebra, through a journey from the natural numbers to prime fields and finite fields, with some detours. Many books are devoted to the construction of these fields from the natural numbers. Perhaps the first one is [Landau 1930], see also the books [Ebbinghaus et.al. 1991], [Lay 2005], and the references therein. An important step in this process is the construction of the field of rational numbers. It turns out that this field is the only (in an appropriate sense) infinite prime field. Motivated by the reading of [Loonstra 1972], I decided to write an essentially self-contained book devoted to the construction not only of the field of rational numbers but of all prime fields and, more interestingly, to the proof of existence and "uniqueness" of all finite fields. Only a knowledge of basic set theory and some familiarity with mathematical reasoning (see for example [11, Chapters 1 and 2]) are assumed.
