## AlgebraThis book comes from the first part of the lecture notes which the author used for a first-year graduate algebra course. The aim of this book is not only to give the students quick access to the basic knowledge of algebra, either for future advancement in the field of algebra, or for general background information, but also to show that algebra is truly a master key or a ?skeleton key? to many mathematical problems. As one knows, the teeth of an ordinary key prevent it from opening all but one door; whereas the skeleton key keeps only the essential parts, allow it to unlock many doors. The author wishes to present this book as an attempt to re-establish the contacts between algebra and other branches of mathematics and sciences. Many examples and exercises are included to illustrate the power of intuitive approaches to algebra. |

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### Contents

Chapter Set theory and Number Theory | 1 |

2 Unique Factorization Theorem | 6 |

3 Congruence | 12 |

4 Chinese Remainder Theorem | 20 |

5 Complex Integers | 23 |

6 Real Numbers and padic Numbers | 33 |

Group theory | 47 |

2 The Transformation Groups on Sets | 55 |

3 Linear Transformation and Matrix | 181 |

4 Module and Module over P I D | 196 |

5 Jordan Canonical Form | 214 |

6 Characteristic Polynomial | 223 |

7 Inner Product and Bilinear form | 232 |

8 Spectral Theory | 243 |

Polynomials in One Variable and Field Theory | 252 |

2 Algebraic Extension | 257 |

3 Subgroups | 62 |

4 Normal Subgroups and Inner Automorphisms | 73 |

5 Automorphism Groups | 82 |

6 pGroups and Sylow Theorems | 85 |

7 JordanHölder Theorem | 89 |

8 Symmetric Group Sn | 96 |

Polynomials | 102 |

2 Polynomial Rings and Quotient Fields | 108 |

3 Unique Factorization Theorem for Polynomials | 114 |

4 Symmetric Polynomial Resultant and Discriminant | 130 |

5 Ideals | 144 |

Linear Algebra | 160 |

2 Basis and Dimension | 165 |

### Common terms and phrases

algebraic algebraic closure associated assume b₁ basis called canonical characteristic claim coefficients conclude condition constructed contradiction Corollary cyclic decomposition defined Definition Discussion distinct easy element equation Example Exercises exists factor Find finite fixed follows from Theorem function fundamental Furthermore Galois given group G homomorphism ideal identity integer integral domain invertible irreducible isomorphism Lemma Let G Let us consider linear transformation linearly independent matrix maximal minimal polynomial module multiplication namely normal subgroup Note operation orbits points polynomial polynomial f(x positive integer preceding prime prime number primitive Proof Prove rational real numbers relation respect ring root rotation satisfies separable sequence Show sides solution solved splitting field subgroup subgroup of G subset Suppose Theorem theory unique unit usual variables vector space