数论元素（True PDF）
This book is intended to complement my Elements oi Algebra, and it is similarly motivated by the problem of solving polynomial equations. However, it is independent of the algebra book, and probably easier. In Elements oi Algebra we sought solution by radicals, and this led to the concepts of fields and groups and their fusion in the celebrated theory of Galois. In the present book we seek integer solutions, and this leads to the concepts of rings and ideals which merge in the equally celebrated theory of ideals due to Kummer and Dedekind. Solving equations in integers is the central problem of number theory, so this book is truly a number theory book, with most of the results found in standard number theory courses. However, numbers are best understood through their algebraic structure, and the necessary algebraic concepts rings and ideals-have no better motivation than number theory. The first nontrivial examples of rings appear in the number theory of Euler and Gauss. The concept of ideal-today as routine in ring the ory as the concept of normal subgroup is in group theory-also emerged from number theory, and in quite heroic fashion. Faced with failure of unique prime factorization in the arithmetic of certain generalized "inte gers" , Kummer created in the 1840s a new kind of number to overcome the difficulty. He called them "ideal numbers" because he did not know exactly what they were, though he knew how they behaved.
English | PDF（True）| 2003 | 266页| ISBN:0387955879 | 27.3 MB这本书旨在补充我的《代数基础》，它同样受到求解多项式方程问题的启发。然而，它独立于代数书，而且可能更容易。在《代数元素》中，我们通过部首寻求解，这导致了域和群的概念及其在著名的伽罗瓦理论中的融合。在本书中，我们寻求整数解，这导致了环和理想的概念，这些概念融合在Kummer和Dedekind同样著名的理想理论中。整数方程组的求解是数论的核心问题，因此这本书是一本真正的数论书，大部分结果都可以在标准数论课程中找到。然而，数字最好通过它们的代数结构来理解，必要的代数概念——环和理想——没有比数论更好的动机。环的第一个重要例子出现在欧拉和高斯的数论中。今天，理想的概念作为循环理论中的常规，就像正规子群的概念在群论中一样，也是从数论中出现的，而且是以相当英勇的方式出现的。面对某些广义“整数”算法中唯一素数因式分解的失败，Kummer在19世纪40年代创造了一种新的数字来克服这一困难。他称之为“理想数字”，因为他并不确切知道它们是什么，尽管他知道它们的行为方式。本站不对文件进行储存，仅提供文件链接，请自行下载，本站不对文件内容负责，请自行判断文件是否安全，如发现文件有侵权行为，请联系管理员删除。
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