## A Course on Abstract AlgebraThis textbook provides an introduction to abstract algebra for advanced undergraduate students. Based on the author's lecture notes at the Department of Mathematics, National Chung Cheng University of Taiwan, it begins with a description of the algebraic structures of the ring and field of rational numbers. Abstract groups are then introduced. Technical results such as Lagrange's Theorem and Sylow's Theorems follow as applications of group theory. Ring theory forms the second part of abstract algebra, with the ring of polynomials and the matrix ring as basic examples. The general theory of ideals as well as maximal ideals in the rings of polynomials over the rational numbers are also discussed. The final part of the book focuses on field theory, field extensions and then Galois theory to illustrate the correspondence between the Galois groups and field extensions. This textbook is more accessible and less ambitious than most existing books covering the same subject. Readers will also find the pedagogical material very useful in enhancing the teaching and learning of abstract algebra. |

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### Contents

1 Preliminaries | 1 |

2 Algebraic Structure of Numbers | 17 |

3 Basic Notions of Groups | 35 |

4 Cyclic Groups | 53 |

5 Permutation Groups | 65 |

6 Counting Theorems | 81 |

7 Group Homomorphisms | 95 |

8 The Quotient Group | 109 |

13 Ideals and Quotient Rings | 193 |

14 Ring Homomorphisms | 205 |

15 Polynomial Rings | 223 |

16 Factorization | 239 |

17 Vector Spaces Over an Arbitrary Field | 261 |

18 Field Extensions | 273 |

19 All About Roots | 295 |

20 Galois Pairing | 315 |

### Common terms and phrases

additive group automorphism bijective called commutative ring complex numbers conjugate construct Corollary cyclic group Define denote element of order epimorphism equivalence relation Example factor field extension field F field of f(x Find finite abelian group finite extension finite group follows function G contains G F[x g G G group G group homomorphism group of order Hence integral domain irreducible over Q irreducible polynomial Lemma Let F Let f(x Let G Let H linear matrix maximal ideal minimal polynomial nonzero normal subgroup one-to-one permutation polynomial of degree polynomial ring positive integer positive prime integer Proof Proposition prove quotient field rational numbers relatively prime ring homomorphism ring with identity sending Show that G space over F splitting field subfield subgroup of G subgroup of order subnormal series subring subset Suppose Sylow p-subgroup unique vector space