## 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

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 JordanHolder 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 |

3 Algebraic Closure | 271 |

4 Characteristic and Finite Field | 274 |

5 Separable Algebraic Extension | 282 |

6 Galois Theory | 291 |

7 Solve Equation by Radicals | 306 |

8 Field Polynomial and Field Discriminant | 321 |

9 Liiroths Theorem | 326 |

Appendix | 332 |

A2 Peanos Axioms | 333 |

A3 Homological Algebra | 337 |

343 | |

### Common terms and phrases

algebraic closure algebraic extension basis canonical map characteristic values claim coefficients commutative group commutative ring complex integers complex numbers composition series conclude constructed Corollary Discussion easy Example factor Find finite extension finite field following definition following equation following theorem follows from Theorem function fundamental sequence Galois extension Galois group group G group of order homomorphism ideal integral domain invertible irreducible elements irreducible polynomial isomorphism Jordan canonical form Let G Let us consider Let us define linear transformation linearly independent linearly independent set matrix form maximal minimal polynomial module natural numbers Noetherian non•trivial non•zero normal series normal subgroup Note orbits permutation polynomial f(x positive integer prime decomposition prime number primitive polynomial Proof Prove quotient groups real numbers root of unity rotation self•adjoint separable algebraic Show splitting field subgroup of G subset subspace Suppose surjective transformation group unique variables vector space Zorn's Lemma