wenweili/aljabr-1

Methods in Algebra (Volume 1): A Chinese textbook on Algebra

This is the LaTeX source for the textbook Methods of Algebra (in Chinese: 代数学方法), Volume 1, revised version (February 2023).

The book is originally published in 2019 by Higher Education Press (Beijing), ISBN 978-7-04-050725-6. The revised version has been published.

The PDF version and the errata are available on the author's web page. The contents in this revised version are "frozen": the spotted mistakes will be corrected in the next revision/edition.

It is hoped that these files will be of some help to those professors, students as well as amateurs who wish to write serious Chinese books in Mathematics or Physics without too much TeXnical trouble.

How to compile

System requirements

The files are to be compiled using XeLaTeX with the xeCJK package. The reader is assumed to work under the UN*X + bash environment.

The recipes below can be tweaked to work under Windows, but this is not recommended. The simplest solution is to go open-source.

We only need the standard packages and fonts, such as

• TeX Live, including the programs latexmk, xindy and biber.
• Standard fonts included in TeX Live, in particular the Fandol fonts. For some strange reason I used and installed the TeX Gyre Heros fonts in the system. In case of error messages related to these fonts, please look for the OTF files (inside the directories in your computer which store TeX-related fonts) whose names start with texgyreheros, and install them manually in your system.
• The Noto Sans CJK SC fonts from Noto CJK, which should also be installed system-wide.

Make sure that all the relevant packages/programs are installed. For reference, the author made the compilation using Arch-based Linux distributions with TeX Live 2018 and/or 2020; the packages biber and texlive-science are required.

Clone the files

Assume that Git has been installed on your computer. As a preparation for the compilation process, we will clone the files into ~/AlJabr-1 in our home directory. In command line, type

cd ~
git clone https://github.com/wenweili/AlJabr-1

All the source files are encoded in UTF-8, which is the de facto standard for storing multilingual texts. If you encounter problems in opening the source files under Windows, try to re-configure your editor or convert the encoding manually.

Compile the TeX source

Move to the directory

cd ~/AlJabr-1

Then, either type

latexmk -pdf -pdflatex="xelatex -shell-escape -interaction=nonstopmode %O %S" Al-jabr-1

under bash, or more simply

make

Have a cup of coffee since this will take several minutes. The resulting PDF file should appear as Al-jabr-1.pdf in the same directory. Note that the main file is Al-jabr-1.tex.

To clean up everything in our directory except the PDF file, type

make clean

The document class AJbook

The book is written in the AJbook class (AJbook.cls). This is a general-purpose document class, based on XeLaTeX/xeCJK and the standard book class in LaTeX, whose aim is to produce Chinese books in Mathematics/Physics of professional quality. Its basic usage is illustrated in Template-AJbook.tex; type

latexmk -pdf -pdflatex="xelatex -shell-escape -interaction=nonstopmode %O %S" Template-AJbook

or more simply

make template

to compile the template.

The fonts and other apppearances are customizable through several config files; please read the source files carefully for further details. The Template-AJbook.tex file follows the original configurations of the book.

The errata

The errata is produced from Errata-Al-jabr-1.tex, which is based on the really simple document class file AJerrata.cls. Apart from the standard fonts bundled with TeX, it also depends on Noto Serif CJK SC and Noto Sans CJK SC; you can install them from Noto CJK fonts.

To compile the errata, type

xelatex Errata-Al-jabr-1

or

make errata

in the same directory.

The errata to the previous editions are named as Errata-Al-jabr-1-v0.tex, and so forth.

Feedback

In case of problems of compilation, please kindly report to the author. Make sure that all the system requirements above are met, and provide detailed error messages. Other suggestions are also welcome.

License

The entire codebase is under CC BY 4.0.

aljabr-1's Issues

堪误

AlJabr-1/chapter1.tex

Lines 191 to 195 in 881cc79

	\begin{example}[无穷序数 $\omega$ 的构造] \index[sym1]{omega@$\omega$}
	考虑序数 $0 := \emptyset$, 不断取后继得到序数
	\[ 1 := 0 + 1, \quad 2 := 1+1, \quad 3=2+1, \ldots \]
	等等. 现在我们着手定义最小的非零极限序数 $\omega$, 这样的序数如存在则必包含所有 $0,1,2, \ldots$, 而且实由它们构成. 首务是证明非零极限序数存在: 回忆无穷公理 \textbf{A.6} 中归纳集的概念, 若 $x$ 是归纳集, 取 $\alpha := \{y \in x : y \subset x, \; y \in \textbf{On} \}$. 按定义直接验证以下性质:
	\begin{inparaenum}[(a)]

193行的 3 = 2 + 1 应为 3 := 2 + 1

感谢李老师开源的书籍和Latex模板

感谢李老师开源的书籍和Latex模板，我喜欢这套Latex模板，把这套模板搬运过去单开了一个项目（https://github.com/lichuang/latex-template），以后自己记的Latex笔记都可以用上这套模板，后续也可以根据自己需要去二次定制一下。

关于维数良定义性: 一个不区分有限维与无限维的证明.

我希望探讨在维数良定义性的证明中不区分有限与无限维情形的可能性.
(理想情况下应当同时适用于向量空间与超越扩张的情形.)

(动机: 我认为当前书中使用基数运算的证明有偏离"代数"主线之嫌.)

欲证明任意无关子集 S 的基数不超过任意张成集 B 的基数.
(以下是我的尝试:)

令 B 是一个张成集, S 是一个无关子集. 称三元对 (B', j, S') 为一个代换 , 若其中 B' 为 B 子集, S' 为 S 子集, j : B' \to S' 为一双射, 使得 (S \setminus S') \sqcup B' 无关.

空代换 B' = \emptyset = S' 的存在性由 S 的无关性得出. 又由于相关性只涉及有限多个元素, Zorn 引理适用, 故存在一个极大代换 (B', j, S').

兹断言 S' = S. 否则选取 s \in S \setminus S', 则任意 b \in B \seminus B' 都与 \big( S \setminus (S' \sqcup {s}) \big) \sqcup B' 相关. (不然, 延拓 j(b) := s, 但这与给定代换的极大性矛盾.)

于是此时 \big( S \setminus (S' \sqcup {s}) \big) \sqcup B' 的张成已然包含张成集 B, 故 s 须与之相关, 但这与 (B', j, S') 是一个代换矛盾.

故所断言 S' = S 成立, 即得 |S| = |S'| = |B'| \leq |B|, q.e.d.

(视上下文将 "相关/无关" 解释为 "线性~" 或 "代数~", 后者的 "张成" 应理解为 "诸元素所生成子域的在给定大域中的代数闭")

请问这个论证 (若正确) 是否适宜取代当前书中关于维数良定义性的证明?

(这或许会完全取代拟阵在此书中的角色. 但另一方面, 即使 B 与 S 均为基底, 也不能保证极大代换中的 j 必然是双射, 故也不能保证 (B \setminus B') \sqcup S 是基底, 所以这并不能完全取代换元性质: 他们大约相差一个 Bernstein 定理的距离.)

或者是否可以考虑推广拟阵的定义, 使得这个证明可以抽象地适用于 "广义拟阵"?

定义4.4.6不动点集似未出现在符号索引表中

（2023.2网络版）p14的 $Y$ 到 $X$ 的所有映射 $X^Y$ 和p338的不动域 $E^H$均有索引。而p115的不动点集 $X^M$ 与二者均不相同，似应该另加索引。

勘误p264

AlJabr-1/chapter7.tex

Line 173 in 0bbc4b5

\item $x$ 包含于 $A$ 的一个忠实 $R[x]$-子模 $M$, 其中 $M$ 作为 $R$-模有限生成.

\item $x$ 包含于 $A$ 的一个忠实 $R[x]$-子模 $M$, 其中 $M$ 作为 $R$-模有限生成
其中$x\in M$的条件是否可以删去？

Page 187(电子版) 勘误

定理5.7.9, 惯性情形的下一行, \mathfrak{p}=x+y\sqrt{-1}\in\mathbb{Z}[-1] 应为 \mathfrak{p}=x+y\sqrt{-1}\in\mathbb{Z}[\sqrt{-1}]

P325 勘误

映至 X^(p^m) - a 在 F 的代数闭包中的唯一根 a^(-p^m)

应为 a^(p^(-m))

Terminology: derived subgroup or commutator subgroup

在阅读到 2. 范畴论基础 中的 2.2 函子与自然变换 中 例 2.2.4 “数学中用到的函子说之不尽, 略举数端如下.” 之 4 时遇到了一点小障碍。

4. 对于任意群 G, 定义导出子群 G der 为子集 {xyx −1 y −1 : x,y ∈ G} 生成的正规
   子群. 商群 G/G der 是交换群, 称作 G 的 Abel 化 (参看引理 4.7.3). 对于任意
   群同态 φ : G → H, 从定义可看出 φ(G der ) ⊂ H der , 因此 φ 诱导出交换群的同
   态 ¯ φ : G/G der → H/H der . 容易验证 G 7→ G/G der , φ 7→ ¯ φ 定义了 Abel 化函子
   Grp → Ab. Abel 化函子不是忠实函子.

读到第一句 “子群 G der” 的时候好奇为何有这样的构造，且商群 G/G der 恰好是交换群的特性更让我思考如此特别的子群定有一个名字来描述它。但指引参看的引理 4.7.3 中没有特别提及什么新的名词。后来搜索了好一番才注意到我断句错了，应该是 “导出子群 G der”。

我在论坛和论文检索中注意到，似乎比起 “导出子群” 能找到的更多的说法有 “导群” 和 “换位子群”，后两者的词法不那么容易令人误读。目前还对代数不甚了解，斗胆提议老师，调整一下此处的术语用法会否更好。

Erratum: Commutative Diagram on Page 57

在第57页下方

(Δ/β) 态射: ........

右侧的交换图表中 应为φ:L'->L
原书所示将导致L为C中的始对象