Abstract Algebra I : basic concepts

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This note is mainly based on "抽象代数 I 代数学基础 孟道骥"

Symbol Assumptions

follow the mainstream definition.

is a field.

1 Basic Concepts

1.1

direct product: Mapping is called if and are sets, and , there exists(and only exists) a element which maps . Let , and we call is the image of under , and is one of the inverse image of .

(等价类): , is the relation.

(商集合):

(自然映射): , mapping is called nature mapping from to

(同余关系): if , then ~ is called (同余关系). is called (同余类).