Abstract Algebra

这里是我的抽象代数笔记。对我本人来说,使用英文记笔记是一次不小的挑战(虽然大部分都是抄书QAQ,但是也在不少地方尝试改成自己的语言,部分证明使用的是自己的方法)。这段话是在本笔记写完第一章之后写的,感觉不知不觉之间就已经记了不少笔记了,在这段时间内我的数学水平也长进了不少,对数学也有了全新的认知。总之,希望你能喜欢我的笔记(虽然没有人读)。

以下是原文:

Here is my personal note of the book, Advanced Modern Algebra, written by Joseph J. Rotman. It's the second time I try to write a note concerning abstract algebra. Maybe this article contains many faults, and if you find some of them, please leave a comment to let me know. Please note that some of the math marks in this article do not use the same mark as in the book, because of the habit of mine.

Abstract algebra is an important part of modern mathematics. It is the study of algebra structures, such as groups, rings and fields etc. It concentrates on the common properties of algebra structures instead of certain numbers or equations. It's the basis for learning many fields of mathematics.

Change log:

  • 2023.6.19: Update the section 1.9.
  • 2023.6.9: Update the section 1.8.
  • 2023.5.24: Update the section 1.7.
  • 2023.5.10: Update the section 1.6.
  • 2023.4.21: Finished the section 1.5 and fixed some bugs.
  • 2023.4.17: Start to write the section 1.5.

Contents:

  1. Commutative Rings
    1.1 Polynomials
    1.2 Homomorphisms
    1.3 Quotient Rings
    1.4 From Arithmetic to Polynomials
    1.5 Maximal Ideals and Prime Ideals
    1.6 Finite Fields
    1.7 Irreducibility
    1.8 Euclidean Rings and Principal Ideal Domains
    1.9 Unique Factorization
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