Abstract Algebra I
Nithi Rungtanapirom
This is not a book; it is a published 59-page abstract algebra articleContents
0.Prerequisites in Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
I.Groups5
1.Basic Concepts of Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
2.Powers and Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8
3.Cosets and Quotient groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11
4.Group Actions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
5.The Symmetric Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20
6.Products of groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22
7.Sylow Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25
8.Structure Theorem for Finite Abelian Groups. . . . . . . . . . . . . . . . . . . .28
II. Rings and fields31
9.Basic Concepts of Rings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31
10.Ideals and Quotients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35
11.Constructions of Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41
12.Fundamentals of Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45
13.Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50
14.Field extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53
Bibliography59
0.Prerequisites in Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
I.Groups5
1.Basic Concepts of Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
2.Powers and Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8
3.Cosets and Quotient groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11
4.Group Actions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
5.The Symmetric Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20
6.Products of groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22
7.Sylow Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25
8.Structure Theorem for Finite Abelian Groups. . . . . . . . . . . . . . . . . . . .28
II. Rings and fields31
9.Basic Concepts of Rings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31
10.Ideals and Quotients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35
11.Constructions of Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41
12.Fundamentals of Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45
13.Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50
14.Field extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53
Bibliography59
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